Geometry: A Complete Course - VideoText · examples are considered,in noticeable detail, taking students, once again, through the logic of the concept development process.The premise

Written by: Larry E. Collins

Geometry:A Complete Course

(with Trigonometry)

Module B – Instructor's Guidewith Detailed Solutions for

Progress Tests

RobbinsCreative

Errata March 2015

Geometry: A Complete Course (with Trigonometry)Module B - Instructor's Guide with Detailed Solutions for Progress Tests

Copyright © 2014 by VideotextInteractive

Send all inquiries to:VideotextInteractiveP.O. Box 19761Indianapolis, IN 46219

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted,in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the priorpermission of the publisher, Printed in the United States of America.

ISBN 1-59676-098-21 2 3 4 5 6 7 8 9 10 - RPInc - 18 17 16 15 14

Letter from the author . . .Welcome to Module B of the VideoText Interactive Geometry Course. I hope you had good success inModule A, and I certainly hope you contacted us if you had questions. Please remember, we are here tohelp. And don't forget that a lot of your questions may be addressed in the Program Overview and theScope and Sequence Rationale in the pages following the Table of Contents.

As was stated in numerous lessons in Module A, you have been preparing to “play the game of PlaneGeometry”. To begin with, you have spent a considerable amount of time revisiting familiar areas of thesubject and investigating the reasons behind the terms and rules you have learned in the past. As well, I'msure you have encountered many new concepts, not only in the discovery of different types of Geometry,but also in our brief study of Mathematical Deductive Logic. You must remember that a general mastery ofthese concepts is essential, if you are to be able to understand the relationships in Plane Geometry, and beable to formally prove those relationships.

In this Module then, we will begin developing the “rules of the game”, by investigating the essentialelements of Plane Geometry, based on the work done by the renowned mathematician, Euclid. Theelements introduced will include Fundamental Terms, both those which will be accepted without definition(Undefined Terms) and those which will be clearly and completely explained, according to the formal rulesof logic (Defined Terms). Further, we will introduce the seemingly obvious relationships between theFundamental Terms, and which we will accept as valid, without proof (Postulates or Axioms).

Of course, this means that, when we have completed the module, we will have determined the basic rulesthat will guide our study. We will then be ready to prove conclusively, in our next module, the fundamentalrelationships (Theorems) we will use to explore the simple closed plane curves that make up PlaneGeometry.

Again, as in Unit I, to make sure that every concept is mastered, and internalized, it will be essential thatyour student do every exercise in the Student WorkText. We want to do everything we can to cover theseconcepts from every perspective possible.

Thank you again for your continued interest in the VideoText Interactive mathematics programs, and goodluck as you now formally begin the study of Plane Geometry. Of course, as always, call us on the help-line,should you need some assistance.

Thomas E. Clark, Author

Table of ContentsInstructional Aids

Program Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iiiScope and Sequence Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v

Detailed Solutions for Progress TestsUnit II - Fundamental TermsPart A - Undefined Terms

LESSON 1 - In AlgebraLESSON 2 - In Geometry

Quiz A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Quiz B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

Part B - Defined TermsLESSON 1 - Good Definitions

Quiz A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Quiz B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

LESSON 2 - Definitions about PointsQuiz A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13Quiz B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

LESSON 3 - Definitions about LinesLESSON 4 - Definitions about RaysLESSON 5 - Definitions about Line Segments

Quiz A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17Quiz B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

LESSON 6 - Definitions about Angles as Sets of PointsLESSON 7 - Definitions about Measurements of Angles

Quiz A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23Quiz B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

LESSON 8 - Definitions about Pairs of AnglesQuiz A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27Quiz B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

LESSON 9 - Definitions about CirclesQuiz A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31Quiz B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33

Module A - Table of Contents i

combined

combined

combined

Part C - Postulates (or Axioms)LESSON 1 - Need

Quiz A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37Quiz B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39

LESSON 2 - Postulate 1 - Existence of PointsLESSON 3 - Postulate 2 - Uniqueness of Lines, Planes, and SpacesLESSON 4 - Postulate 3 - One, Two, and Three Dimensions

Quiz A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Quiz B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

LESSON 4 - Postulate 3 - One, Two, and Three DimensionsLESSON 5 - Postulate 4 - Separation of Lines, Planes, and SpacesLESSON 6 - Postulate 5 - Intersection of Lines or Planes

Quiz A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Quiz B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

LESSON 7 - Postulate 6 - RulerLESSON 8 - Postulate 7 - Protractor

Quiz A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Quiz B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

LESSON 9 - Postulate 8 - CircleLESSON 10 - Postulate 9 - Parallel LinesLESSON 11 - Postulate 10 - Perpendicular Lines

Quiz A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Quiz B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Unit II Test - Form A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Unit II Test - Form B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Unit II Test - Form C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Unit II Test - Form D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

ii Module A - Table of Contents

combined

combined

combined

combined

Program OverviewThe VideoTextInteractive Geometry program addresses two of the most important aspects ofmathematics instruction. First, the inquiry-based video format contributes to the engaging ofstudents more personally in the concept development process. Through the frequent use of the pausebutton, you, as the instructor, can virtually require interaction and dialogue on the part of your student.As well, students who work on their own, can “simulate” having an instructor present by pausing thelesson every time a question is asked, and trying to answer it correctly before continuing. Of course, thestudent may answer incorrectly, but the narrator will be sure to give the right answer when the playbutton is pressed to resume the lesson. Right or wrong, however, the student is regularly engaging inanalytical and critical thinking, and that is a healthy exercise, in and of itself. Second, eachincremental concept is explored in detail, using no shortcuts, tricks, rules, or formulas, and no stepin the process is ignored. As such, the logic and the continuity of the development assure students thatthey understand completely. Subsequently, learning is more efficient, and all of the required concepts(topics) of the subject can be covered with mastery. Of course, the benefits of these efforts can be seeneven more clearly in a description of a typical session, as follows:

After a brief 2 or 3 sentence introduction of the concept to be considered, usually by examining thedescription, and the objective given at the beginning of the video lesson, you and your student can begin.You should pause the lesson frequently, usually every 15-20 seconds (or more often if appropriate), toengage your student in discussion. This means that, for a 5-10 minute VideoText lesson, it may take 10-15 minutes to finish developing the concept. Dialogue is a cornerstone. In addition, during this time,your student should probably not be allowed to take notes. Students should not have their attentiondivided, or they risk missing important links. Neither should you be dividing your attention, by lookingat notes, or writing on a pad, or an overhead projector. Everyone should be concentrating on conceptdevelopment and understanding. Please understand that a student who is accustomed to workingalone, or can be motivated to study independently, has, with the VideoText, a powerful resource toexplore and master mathematical concepts by simulating the dialogue normally encountered with a“live” instructor. And, because of the extensive detail of the explanations, along with the computergenerated graphics, and animation, students are never shortchanged when it comes to the insightnecessary to fully comprehend.

Once the concept is developed, and the VideoText lesson is completed, you can then employ the CourseNotes to review, reinforce, or to check on your student's comprehension. These Course Notes arereplications of the essential content that was viewed in the VideoText lesson, illustrating the same terms,diagrams, problems, numbers, and logical sequences. In fact, at this time, if your student needs a littlemore help, he or she can use the Course Notes while viewing the lesson again, using them as a guide, tore-examine the concept. The key here is that students concentrate on understanding first, and takecare of documentation later.

Please understand that it is not the intent of the program to let the VideoText lesson completely take theplace of personal instruction or interaction. Actually, the video should never tell your studentsanything that hasn't been considered or discussed (while the lesson is paused), and it should neveranswer questions that have not already been considered and resolved. As such, it becomes a “newbreed” of chalkboard or overhead projector, whereby you, as the teacher, or your student working alone,can “write”, simply by pressing the “play” button. This is a critical point to be understood, and should

iii

serve to help you examine all of the materials and strategies from the proper perspective.Next, your student can begin to do some work independently, either by your personal introduction ofadditional examples from the WorkText, or by the student immediately going to the WorkText on his orher own. The primary feature of the WorkText, beside providing problem banks with which studentscan work on mastery, is that objectives are restated, important terms are reviewed, and additionalexamples are considered, in noticeable detail, taking students, once again, through the logic of theconcept development process. The premise here is simple. When students work with an instructor,whether doing exercises on their own, or working through them with other students, they are usuallyconcentrating more on “how to do” the problems. Then, when they leave the instructor, they simplydon't take the discussion of the concept with them. The goal of this program is to provide a resourcewhich will help students “re-live” the concept development on their own, whether for review, or foradditional help. That is the focus of the Student WorkText.

Having completed the exercises for the lesson being considered, your student is now ready to use thedetail in the Solutions Manual to check work and engage in error analysis. Again, it is essential to astudent's understanding that he or she find mistakes, correct them, and be required to give someexplanation, either verbal or in writing, to you as the instructor. In fact, at this stage, you might evenconsider grading your student only on the completion of the work, not on its accuracy. Remember,this is the first time the student has tried to demonstrate understanding of a concept, and he or she maystill need some fine-tuning. So, because this is part of the initial learning process, the focus should beon a careful analysis of the logic behind the work, not just the answers. Finally, it is time to assessyour student's mastery of the concept behind the work. Just be sure you are not testing on the sameday the exercises were completed. Short-term memory can trick you into thinking that you “have it”,when, in fact, you are just remembering what you did moments before. A more accurate evaluation canbe made on the next day, before moving on to the next lesson. Further, the quizzes and tests in theprogram often utilize open-response questions which will require your student to state, in writing,his or her understanding of the concept. This often reveals much more about a student'sunderstanding than just checking to see if an answer on a test is correct. Remember too, that there aretwo versions of every quiz and test, allowing you to retest, if necessary, in order to make sure that yourstudent has mastered the concept.

Of course, just as with the WorkText, there are detailed solutions for all of the quiz and test problems,in the Instructor's Guide. Again, your student should be required to analyze problems that weremissed, and explain why the problem should have been done differently. It is simply a fact that one ofthe most powerful and effective teaching tools you can employ, is to ask your students to “articulate”to you what their thinking was, as they worked toward a given answer.

As you can see, the highly interactive quality of this program, affords students a much greateropportunity than usual to grow mathematically, at a personal level, and develop confidence in theirability. That can have a tremendous impact on a student's future pursuits, especially in an age whereapplications of mathematics are so important.

iv

Scope and Sequence RationaleThere are two basic premises which drive concept development in Geometry, and these two essentialsshape the logical scope and sequence of geometric content.

First, it is generally understood that Geometry is the study of spatial relations. In the same way thatAlgebra is the study of numerical relations (equations and inequalities), and Calculus is concernedprimarily with rates of change, Geometry is a comprehensive exploration of “shapes” (as sets of points),the measurements associated with those shapes, and the relationships that can be established betweenthose shapes. As such, no treatment of Geometry should ever investigate those relationships onlyindividually, or in isolation. This is especially noticeable with traditional textbooks, which generally usea format which addresses them in different “chapters”. In the VideoText Interactive Geometry course,concepts are discussed from a “Unit” perspective, pursuing and connecting, in an exhaustive way,all of the outcomes associated with various possibilities for a specific relationship. Of course, asmuch as is possible, students need to “see” those relationships, and experience the “motion”, or“transformation”, necessary to clearly illustrate the concept. It really is impossible to put a value on thebenefits of visualization, in life in general, and in Geometry in particular. So, in the VideoTextInteractive Geometry program, computer-generated graphics are used extensively, along withanimation and color-sequencing, in order that students can actually see the relationships develop.

The second premise is that geometric concepts should be studied utilizing all of the power andconviction that both inductive and deductive reasoning can bring to the table. In other words, it isalways desirable, and helpful, for students to “experiment”, inductively, with a geometric relationship, inan effort to come to some general conclusion. Once that general conclusion has been arrived at,however, it is even more convincing if the student is able to “prove”, deductively, that the conclusionabsolutely must follow, logically, from the given information. No, formal proof is not often asked for ineveryday life. On the other hand, the exercise of developing that kind of thinking is invaluable, not onlyin some specific job-related activities, but, more generally, in the daily problem-solving situations thatconfront us. The VideoText Interactive Geometry program is formatted in such a way that formal proofis a cornerstone.

Unit I, then, focuses on a complete preparation for students to begin a formal study of Geometry by“re-teaching” of all of the basic geometric concepts for which students have simply memorized theappropriate term, definition, or formula. That means we must re-establish that Mathematics in general,and Geometry in particular, is a language, with parts of speech and sentence structure. We mustdevelop, in detail, the concepts associated with building geometric shapes. We must investigate, againin detail, the concepts dealing with the measurement of those shapes. Finally, we must thoroughlydevelop the principles of inductive and deductive reasoning, giving significant attention to the dynamicsof mathematical deductive logic, which are the building blocks that students will use to construct formalproofs.

In Unit II, we begin the actual study of “Plane Geometry” by developing all of the necessary terms,definitions, and assumptions we will be using as a basis for studying geometric relationships. Inother words, we draw on the analogy that studying any area of Mathematics is like “playing a game”.We must first determine which basic elements will be “undefined” in our Geometry, or accepted

v

without definition. We must then determine which basic elements can be formally defined, using thoseundefined terms. Finally, we must build a list of “postulates”, or conditional assumptions which willserve as the “rules of the game”, guiding us through the investigation of relationships, in our Geometry.It is important to note, at this point, that every Plane Geometry study will, in certain ways, be unique tothe philosophy of the instructor, depending on the acceptance of these fundamental terms. In otherwords, while the prevailing context will always be that of classical Euclidean Geometry, the lists ofdefinitions and postulates may differ from person to person. The key, however, is that each study willrely on its own particular list of Essential Elements to prove the rest of the relationships to beinvestigated.

So, in Unit III, we use the Fundamental Terms developed in Unit II, to prove FundamentalTheorems related to points, lines, rays, segments, and angles. These theorems will be foundational tothe study of Simple Closed Plane Curves, which are the primary backdrop of all studies of PlaneGeometry.

At this point, since we have put in place the “rules of the game”, we can begin, and, for all practicalpurposes, complete, a methodic investigation of the geometric relationships associated with Triangles(Unit IV), Other Polygons (Unit V), and Circles (Unit VI). That then allows us to conclude our studyby the investigation of several applications, internal to the study of Geometry.

First, in Unit VII, we will engage in the classic geometric exploration of “Construction”. Thismeans that, with the use of only a straight edge (to construct lines, rays, and segments), and a compass(to construct circles, and arcs of circles), we will attempt to use our knowledge of geometric relationshipsto “build”, and “operate on”, various geometric shapes. Included will be the replication and division ofline segments and angles, the building of polygons to desired specifications, and the generation of circlesto desired specifications.

Second, in Unit VIII, we will examine, in significant detail, the relationships between the variouscomponents of triangles. This is, of course, the study of Trigonometry, from the Greek, meaning “tri-angle-measure”. Included are the basic relationships of sine, cosine, and tangent, as well asapplications involving the Pythagorean Theorem, the Laws of Sines and Cosines, and several otherambiguous cases.

Please understand that the organizational argument presented here is not meant to stifle the creativity of the instructor. Neither should it prohibit the instructor from utilizing a modular approach to conceptdevelopment. It does, however, serve to remedy the fragmented, isolated topic, “chapter” approach to a subject which has been traditionally presented to us in “textbooks”, without that element ofdevelopmental continuity. To that end, it speaks loudly to the curricular issues which all instructors face, and the attitudinal issues students deal with when they are presented with a new and differentMathematics course.

vi

Name

Class Date Score

Quiz Form B

Unit II - Fundamental TermsPart A - Undefined TermsLesson 1 - In AlgebraLesson 2 - In Geometry

From the list of five mathematical parts of speech and four types of mathematical expressions, identify each of theitems in problems 1 through 9 below, using the most appropriate term.

1. 7n + 5

2. The fraction bar in

3.

4.

5. 2x – 3 = 7

6. The fraction bar in

7. 2 + 9 = 5 ÷ 2

8. The “m” in 2m + 5

9. (8)(3) + 15

© 2014 VideoTextInteractive Geometry: A Complete Course 5

5

≥

5 4 7

2

+ +

5

4

Open Phrase

Technically it is part of the Number Symbol. But it might also be considered a Grouping Symbol or an Operation Symbol.

Number Symbol

Relation Symbol

Open Sentence

Technically it is part of the Number Symbol. But it might also be considered as an Operation Symbol.

Closed Sentence

Placeholder Symbol

Closed Phrase

NameUnit II, Part B, Lesson 2, Quiz Form A—Continued—

3. Refer to the figure at the right to identify each of the

following sets of points as coplanar or non-coplanar.

a) F, G, A, and C

b) D, B, A, and C

c) C, G, A, and B

d) A, C, E, and D

4. For each of the given sets of numbers, which number is between the other two?

a)

b)

© 2014 VideoTextInteractive Geometry: A Complete Course14

PC D

B

l

B C

D

J

KH

G

EF

A C

J

E

DK

GF

BHA

D

FE

C

G BA

D

FE

C

G

BA

DC

O

A B

A B C D A B C D

A B C

B A C

A B C D

B A D

A B C

A B C

A B C

J

1 8 1 3 32

. , . ,( ){ }

. , ,7137

11

3

4

Non-Coplanar

Non-Coplanar

Coplanar

Coplanar

3

.713


NameUnit II, Part B, Lesson 2, Quiz Form B—Continued—

3. Refer to the figure at the right to identify each of the

following sets of points as coplanar or non-coplanar.

a) F, G, E, and C

b) D, F, G, and B

c) A, J, E, and D

d) B, F, D, and E

4. For each of the given sets of numbers, which number is between the other two?

a)

b)

PC D

B

l

C

D

J

G

E

C

J

E

DK

GF

BHA

D

FE

C

G BA

D

FE

C

G

BA

DC

O

B

A B C D A B C D

A B C

B A C

A B C D

B A D

A B C

A B C

A B C

J

2

3,

3

4,

4

5

.615, .636, .513 { }

Non-Coplanar

Coplanar

Coplanar

Coplanar

3

4

.615

Name

Class Date Score

Quiz Form A

Unit II - Fundamental TermsPart B - Defined TermsLesson 3 - Definitions about LinesLesson 4 - Definitions about RaysLesson 5 - Definitions about Line Segments

1. How many endpoints does a line segment have? A ray? A line?

2. Name the nine line segments in the following diagram.

3. How many rays on a line may have a common end point?

4. How many line segments may be found on a line?

5. How many line segments on a line may have a common endpoint?

6. Two line segments lie on the same line. Can we then say that the union of the two line segments must be a line segment? (yes or no) Sketch an illustration for your answer.

7. Two rays lie on the same line. Can we then say that the union of the two rays might be a ray? (yes or no) Sketch an illustration for your answer.

8. Two rays lie on the same line. Can we then say that the intersection of the two rays must be a line segment? (yes or no) Sketch an illustration for your answer.


2, 1, None

AB, BC, AC, AD, DO, DB, AO, OC, BO

2

An infinite number

An infinite number

No, AB U CD = {all points on AB or CD or both}

PC

mk

EF

D

B

n

l

B C

D

J

KH

G

EF

A C

J

E

DK

GF

BHA

E

A

DC

O

A B

A B C D

A B C

B A C

A B C D

B A D

A B C

A B C

PC

mk

EF

D

B

n

l

B C

D

J

KH

G

EF

A C

J

E

DK

GF

BHA

D

FE

C

G BA

D

FE

C

G

BA

DC

O

A B

A B C D A B C D

A B C

B A C

A B C D

B A D

A B C

A B C

A B C

J

PC

mk

EF

D

B

n

l

B C

D

J

KH

G

EF

A C

J

E

DK

GF

BHA

D

FE

C

G BA

D

FE

C

G

BA

DC

O

A B

A B C D A B C D

A B C

B A C

A B C D

B A D

A B C

A B C

A B C

J

PC

mk

EF

D

B

n

l

B C

D

J

KH

G

EF

A C

J

E

DK

GF

BHA

D

FE

C

G BA

D

FE

C

G

BA

DC

O

A B

A B C D A B C D

A B C

B A C

A B C D

B A D

A B C

A B C

A B C

JYes, AB U BC = AC

No, AB AC = A (a point)

U

9. Is it true that the union or intersection of two sets of points on a line must be a subset of the line? (yes or no) Explain your answer.

10. For each of the following, if you are given the coordinates c, d, s, and t, for the points C, D, S, and Trespectively, determine if . Show your work.

c d s t

a) 7 12 –8 –13

b)

c) a – b a + b 3b 5b

11. Find the coordinate of the midpoint C of a line segment, if the coordinates of the endpoints are 5x2 and 13x2.(Hint: Think about a number line)

12. What is meant if we write AB = CD?


Unit II, Part B, Lesson 3,4,5, Quiz Form A—Continued—

Name

CD ST≅

312

−714

−1 934

Yes, even the empty set (if the two sets have no points in common) is a subset of every set.

Yes

Yes

Yes

12 7

13 8

5

−

− − −( )

− −

− −

71

43

1

2

93

41

103

4

( )

a b a b

b b

b

+( ) − −( )−5 3

2

5 13

2

8

24

2 2 22

x x xx

–= =

We mean that AB and CD represent the same measure (number);

that is, the length of AB is the same as the length of CD. Here AB and CD are symbols or names for the

same number.

So, the coordinate of the midpoint C is 5x2 + 4x2 or 9x2

9. What is the basic difference between a ray and each of the following?

a) A line segment

b) A half line

c) A line

10. For each of the following, if you are given the coordinates c, d, s, and t, for the points C, D, S, and Trespectively, determine if ? Show your work.

c d s t

a) 6 9 14 17

b) 4.72 8.35 5.15 9.78

c)


Unit II, Part B, Lesson 3,4,5, Quiz Form B—Continued—

Name

Yes

No

Yes

9 6

17 14

3

−

−

8 35 4 72

3 63

9 78 5 15

4 63

. .

.

. .

.

−

−

9 7 4 7

5 7

175 0

175

25 7

5 7

−

−

⋅

4 7 9 7 0 175

CD ST≅

A ray has only one endpoint, while a line segment has two endpoints.

A ray has one endpoint and a half-line has no endpoint (technically). It may be argued that there is no difference,

since, to draw a half-line, you must start somewhere, but the strict definition of a half-line does not allow for an endpoint

A ray has one endpoint, while a line has no endpoints.

11. AB = 7x – 2, BC = 2x + 8, and B is the midpoint of AC. Find the values of x, AB, BC, and AC.

12. May we write AB = CD? Explain your answer.

Unit II, Part B, Lesson 3,4,5, Quiz Form B—Continued—

Name


7 2 2 87 2 2 2 8 2

7 2 2 10 25 10

x xx x

x x x xxx

− = +− + = + +

− = + −== 22

ABBCAC

= − == + == + =

7 2 2 122 2 8 1212 12 24

( )( )

No. Even if the lengths of AB and CD are equal, you cannot say

that the line segments, as sets of points, are equal. You would have to say that the line segments are congruent (>). In fact, the

only way to use the equality symbol, would be to place a lower-case m in front of the line segments, indicating that the measures

of the line segments are equal (mAB = mCD), or to remove the segments from over the letters, also indicating that the measures

are equal (AB = CD).


Name

Class Date Score

Quiz Form A

Unit II - Fundamental TermsPart B - Defined TermsLesson 6 - Definitions about Angles as Sets of PointsLesson 7 - Definitions about Measurement of Angles

For exercises 1 through 5, refer to the angle at the right.

1. Name the angle in two ways. __________________________

2. Name the two sides of the angle. __________________________

3. Name the vertex of the angle. __________________________

4. Point F is in the __________________________ of the angle.

5. Point G is in the __________________________ of the angle.

For exercises 6 through 9, refer to the figure at the right.

6. How many angles are there, in the figure? __________________________

7. Name the angles in the figure. __________________________

8. Name the vertex of the angle. __________________________

9. Is WV totally in the interior of angle UWX? Explain your answer. __________________________

10. In defining an angle, is it possible that the rays forming the angle might have different endpoints?

11. May a page of a book serve as a model of a half plane? A perfect model of a half-plane? Explain why or why not.

B C

D

J

KH

G

EF

A

A B C D

G

B

F

C

D

G

IH

S

Q

R

N

M

C

D

D

G

B

F

C

DV

X

U

W

G

IH

J

L

K

/BCD; /DCB

CB; CD

Point C

interior

exterior

3

/UWX; /VWX; /VWU

Point W

No; the endpoint of the ray, point W,

is on the angle, not in the interior of the angle.

No

Yes, it may serve as a model, but not as a perfect model. A page of a book is flat, which is a required quality

of a half-plane, but it has 4 edges, and a half-plane has only one edge, defined by the separation line.


12. Use your protractor to measure each angle below. Write your answer, being sure to use correct notation.

a) b)

___________________ ___________________

c) d)

___________________ ___________________

e) f)

___________________ ___________________

13. Name two examples of different size angles in your classroom. ___________________________________

______________________________________________________________________________________

Unit II, Part B, Lesson 6&7, Quiz Form A—Continued—

Name

V

XC

A

B

BE

C

F

D

M

O

N

PR

Q

W

X49o 115o 80o

T

U

P

Q

M

KL

IJ

H

Q

X R

TP

AC

D

B

Q

U T

V

A B X

V

X

U

W

C

A

B

BE

C

F

D

J

L

K

M

O

N

PR

Q

W

X

U

Y

V

49o 115o 80o

120 o 90o

T

U

P

Q

M

KL

IJ

H

Q

X R

TP

AC

D

B

o

F

D

M

PN

Q

MU T

V

A B X

G

B

F

C

DV

X

U

W

C

A

B

EF

D

G

IH

J

L

K

M

O

N

PR

Q

S

Q

R

N

MW

X

U

Y

V

49o 115o 80o

12o120 o 90o

T

U

P

Q

M

KL

IJ

H

Q

X R

TP

AC

D

B

60o60o

A FB E

C D

M

PN

Q

U T

V

A B X

C

D

J

G

E

C D

G

B

F

C

DV

X

U

W

C

A

B

G

IH

J

L

K

M

O

N

S

Q

R

N

MW

X

U

Y

V

49o 115o

12o120 o 90o

T

U

P

Q

M

KL

IJ

H

60o60o

A FB E

C D

M

PN

Q

U T

V

V

XC

A

B

BE

C

F

D

L

M

O

N

PR

Q

W

X49o 115o 80o

U

Q

M

KL

IJ

H

Q

X R

TP

V

X

U

W

C

A

B

BE

C

F

D

J

L

K

M

O

N

PR

Q

W

X

U

Y

V

49o 115o 80o

120 o 90o

T

U

Q

M

KL

IJ

H

Q

X R

TP

m/ABC = 107O m/DEF = 90O

m/GHI = 20O m/JKL = 50O

m/MNO = 60O m/PQR = 165O

(answers will vary) (1) Corner of door forms a

90O angle. (2) Pole holding American flag forms about a 60O angle with the wall.


Name

Class Date Score

Quiz Form B

Unit II - Fundamental TermsPart B - Defined TermsLesson 6 - Definitions about Angles as Sets of PointsLesson 7 - Definitions about Measurement of Angles

1. Suppose that two half-planes in the same plane have the same edge. Describe their union

and intersection.

2. May two consecutive pages of a book serve as a model of a dihedral angle? A perfect model of a dihedral angle? Explain why or why not.


3. Point R is in the __________________________ of angle MNQ.

4. Point Q is __________________________ angle MNQ.

5. Point S is in the __________________________ of angle MNQ.

6. Point N is the __________________________ of angle MNQ.


7. YV forms an angle with ray ________ and ray ________ . (and ray YX)

8. How many angles are in the figure? __________________________

____________________________________________________

9. Name angle WYU another way. __________________________

10. Point V is in the interior of angle UYW and angle __________________________.

11. Point W is in the interior of angle _____________ but in the exterior of angle _____________.

B C

D

J

KH

G

EF

A

G

B

F

C

D

G

IH

S

Q

R

N

M

12o120o

60o60o

A FB E

C D

C

D

D

G

B

F

C

DV

X

U

W

G

IH

J

L

K

S

Q

R

N

MW

X

U

Y

V

12o120 o 90o

Their union is a complete plane, less the edge. Their intersection is empty since the edge is not

included. The edge is not a part of either half-plane.

Yes, it may serve as a model, but not as a perfect model.While both pages

are flat, which is a required quality of half-planes, they both have several edges. A half-plane has only one edge, defined

by the separation line.

interior

on

exterior

vertex

6 ;( , , , , , )∠ ∠ ∠ ∠ ∠ ∠UYX UYV UYW VYW VYX WYX

/UYW

UYX

YU YW

UYX (also VYX) UYV

(/UYX, /UYV, /UYW, /VYW, /VYX. /WYX)


Name

Class Date Score

Quiz Form A

Unit II - Fundamental TermsPart B - Defined TermsLesson 8 - Definitions about Pairs of Angles

Complete each statement in exercises 1 through 6, using the figure at the right.

1. /HIJ and _____________ are vertical angles.

2. /MQL and _____________ are adjacent angles.

3. /JIM and _____________ are supplementary angles.

4. m/MUJ = _____________.

5. The complement of /TUP is _____________.

6. /KLT forms a linear pair with _____________.

In exercises 7 and 8, explain why the angles in each pair are not adjacent.

7. /RXT and /TQP 8. /ABC and /ABD

9. What is the measure of the supplement, of the complement, of a 35 degree angle? (Hint: Make a drawing) ______________________________________________________________________________________

______________________________________________________________________________________

IH

L

K

O

N

P

S

Q

R

N

MW

X

U

Y

V

49o 115o

12o120 o 90o

T

U

P

Q

M

KL

IJ

H

60o60o

A FB E

C D

M

PN

Q

Q R

P

NM

O

U T

V

Q

A

120 o

CB

A

120o

D E

F

X N

Q

M

C

BE

C

F

D

PR

Q

115o 80o

M

Q

X R

TP

AC

D

B

R

A B

W

X

V

X

U

W

C

A

B

BE

C

F

D

J

L

K

M

O

N

PR

Q

W

X

U

Y

V

49o 115o 80o

90o

T

U

P

Q

M

KL

IJ

H

Q

X R

TP

AC

D

B

F

M

PN

Q

Q R

P

NM

O

U T

V

Q T

R

A B

Y Z

W

X

E

F

/TIM or /LIM

/UQL

/TIM or/JIH

90O

/PUM

/TLQ (also /KLJ)

There is no common vertex, and only part of a

common ray.

The common ray, AB, is not between CB and BD.

The complement of 35O is 55O. The supplement of 55 is 125O.


Use the figures above to determine whether each statement in problems 10 through 15 is true or false.

____________ 10. /DEF = 60O

____________ 11. m/ABC = 60O

____________ 12. /ABC = /DEF

____________ 13. m/ABC > m/DEF

____________ 14. m/DEF = 60O

____________ 15. /ABC > /DEF

16. MN bisects /PMQ

m/PMN = 3x + 8

m/QMN = 5x – 10

Find the value of x,and the measure of each angle.

Unit II, Part B, Lesson 8, Quiz Form A—Continued—

Name

12o120 o 90o

T

U

P

Q

M

60o60o

A FB E

C D

M

PN

Q

Q R

P

NM

O

U T

V

120 o

CB

A

120 o

D E

F

X N

Q

M

BCX

WB D

G

IH

J

L

K

M

O

N

PR

Q

S

Q

R

N

MW

X

U

Y

V

49o 115o 80o

12o120o 90o

T

U

P

Q

M

KL

IJ

H

Q

X R

TP

AC

D

B

60o60o

A FB E

C D

M

PN

Q

Q R

P

NM

O

U T

V

Q T

R

A B

Y Z

W

X

120 o

CB

120 o

D E

F

X N

Q

M

False

True

False

False

True

True

3 8 5 10

3 8 3 5 10 3

0 8 2 10

8 10 2

x x

x x x x

x

x

+ = −+ − = − −

+ = −+ = − 110 10

18 2

1

218

1

22

9

+=

⋅ = ⋅

=

x

x

x

m PMN x

m QMN x

∠ = += ( ) += +=

∠ = −= ( ) −

3 8

3 9 8

27 8

35

5 10

5 9 100

45 10

35

= −=


The statements in problems 10 through 15 refer to the figures above. The statements are false. Rewrite eachstatement in the correct form, in the blank to the right.

10. /ABC = 120O ________________________

11. /ABC = /DEF ________________________

12. m/ABC > m/DEF ________________________

13. /DEF > 120O ________________________

14. m/ABC > 120O ________________________

15. /DEF = /ABC ________________________

16. XQ bisects /MXN

m/MXQ = x + 20

m/NXQ = 70

Find the value of x,and the measure of /MXQ.

Unit II, Part B, Lesson 8, Quiz Form B—Continued—

Name60o60o

A FB E

C D

M

PN

Q

Q R

P

NM

O

U T

V

120 o

CB

A

120 o

D E

F

X N

Q

M

S

Q

R

N

MW

X

U

Y

V

49o 115o

12o120o 90o

T

U

P

Q

M

KL

IJ

H

60o60o

A FB E

C D

M

PN

Q

Q R

P

NM

O

U T

V

120o

CB

A

120o

D E

F

X N

Q

M

m/ABC = 120O

/ABC > /DEF

m/ABC = m/DEF

m/DEF = 120O

m/ABC = 120O

/DEF > /ABC or m/DEF = m/ABC

1

220 70

1

220 20 70 20

1

20 50

21

22 50

x

x

x

x

x

+ =

+ − = −

+ =

⋅ = ⋅

== 100

m MXQ x∠ = +

= ( ) +

= +=

1

220

1

2100 20

50 20

70

12


Name

Class Date Score

Quiz Form A

Unit II - Fundamental TermsPart B - Defined TermsLesson 9 - Definitions about Circles

In exercises 1 through 6, refer to the diagram at the right to determine whether the arc is a minor arc, a major arc or a semi circle of circle O.

1.

2.

3.

4.

5.

6.

7. In the figure for exercises 1 through 6, /NOP is called a ____________________ angle.

8. In the figure for exercises 1 through 6, OQ is a ____________________ of the circle, RP is a

____________________ of the circle, point O is the ____________________ of the circle,

and RQ is called a ____________________ of the circle.

9. Tell what fractional part of a circle is represented by the following arcs:

a) an arc of 36O

b) an arc of 160O

10. If an arc is of a circle, it measures ____________________ degrees.

B C

D

J

KH

G

EF

A

A B C D

R

M

QO

PN

P

X

U

6

C

B

A

D

P

(5x)

(x)

o

o

(x)o

C

320

RQ!

MRP"

MNQ"

NP!

RN!

RQP"

minor arc

major arc

semi circle

minor arc

minor arc

semi circle

central

radius

diameter center

chord

36

360

36 1

36 10

1

10= ⋅

⋅= or one tenth-

160

360

40 4

40 9

4

9= ⋅

⋅= or four-ninths

54

3

20

360

1=

3

20

20 18

1= 3 18 = 54o⋅ ⋅ ⋅ ⋅


In exercise 11 through 18, refer to the figure at the right to determine the measure of each arc or angle.

11. m/UPX

12. mXW

13. m/VPU

14. mVX

15. mVW

16. mUVW

17. mUV

18. mVWX

Unit II, Part B, Lesson 9, Quiz Form A—Continued—

Name

C

D

J

G

E

C D

R

M

QO

PN

P

X

U V

W

60

150

E

C

B

A

D

P

(5x)

(x)

o

o

(x)o

B

A

C

(x)o

(2x+30)o

PY

(5x)o

o

o

120O

60O

30O

150O

150O

180O

30O

210O

((

((

((


Name

Class Date Score

Quiz Form B

Unit II - Fundamental TermsPart B - Defined TermsLesson 9 - Definitions about Circles

1. If two arcs of a circle are equal, are their central angles equal? ____________________

2. If an arc of a circle is halved, is its central angle halved? ____________________

In exercises 3 through 10, refer to the diagram at the right.

3. Name four central angles of circle P. _______________________

_______________________

4. Name three minor arcs of circle P. _______________________

_______________________

5. Name two major arcs of circle P. _______________________

_______________________

6. Name a semi circle of circle P. _______________________

7. Name a pair of arcs you think are congruent. _______________________

8. What is mBC? _______________________

9. What is mBG? _______________________

10.What is mCEA? _______________________

R

M

QO

PN

P

X

U V

W

60

150

E

B

C

G

AFH

P

C

B

A

D

P

(5x)

(x)

o

o

(x)o

B

A

C

(x)o

(2x+30)o

P PY

W

U V(3x)o

(8x)o

(8x)o

(5x)o

o

o

100o

60o60o

Yes

Yes

/EPB; /BPC; /CPG; /GPA

GC; CB; BE

AFB

BE BC≅

60O

80O

240O

(other answers possible)




( ( (

EBA; AEC

(( (

(

(

((

(


In exercises 11 through 14, refer to the diagram at the right to find the measures of the indicated arcs.

11. mAD = _________

12. mAB = __________

13. mDAC = __________

14. mCDA = __________


Name

Q

P

P

X

U V

W

60

150

E

B

C

G

AFH

P

A

D

C

B2x+35

3x

2x

x+45

C

B

(5x)

(x)

o

o

B

A

C

(x)o

(2x+30)o

P P (x)

60

Y

M

N

X

PY

W

U V(3x)o

(8x)o

(8x)o

(5x)o

o

o

100o

60o60o

o

o

2 2 35 3 45 360

8 80 360

8 80 80

x x x x

x

x

o+ +( ) + + +( ) =+ =

+ − = 3360 80

8 0 280

8 280

1

88

1

8280

1 35

−+ =

=

⋅ = ⋅

⋅ =

x

x

x

x o

mAD = 2x + 35

= 2 35 + 35

= 70 + 35

= 105o

( )

mAB x

o

== ( )=

2

2 35

70

mDAC mDA mAB mBC

x x x

x

= + += + + + += +=

2 35 2 45

5 80

5 355 80

175 80

255

( ) += +

= o

mCDA mCD mDA

x x

" = += + += ( ) + ( ) +=

3 2 35

3 35 2 35 35

105 ++ +

=

70 35

210o

105O

70O

255O

210O

((

((

mAD = 2x+35

(

mAB = 2x

(

mDAC = mDA + mAB + mBC

( ( ( (

mCDA = mCD + mDA

( ( (

In exercises 15 through 18, refer to circle P to find the value of x. Then find the measure of the labeled arcs.

15. 16.

17. 18.


Name

R

M

QO

PN

P

X

U V

W

60

150

E

B

C

G

AFH

P

A

D

C

B2x+35

3x

2x

x+45

C

B

A

D

P

(5x)

(x)

o

o

(x)o

B

A

C

(x)o

(2x+30)o

P P (x)

60

Y

M

N

X

PY

W

U V(3x)o

(8x)o

(8x)o

(5x)o

o

o

100o

60o60o

o

o

C

D

E

C D

R

M

QO

PN

P

X

U V

W

60

150

E

B

C

G

AFH

P

C

B

A

D

P

(5x)

(x)

o

o

(x)o

B

A

C

(x)o

(2x+30)o

P PY

W

U V(3x)o

(8x)o

(8x)o

(5x)o

o

o

100o

60o60o

E

B

C

G

AFH

P

A

D

C

B2x+35

3x

2x

x+45

C

B

(x)o

(2x+30)o

P (x)

60

Y

M

N

X

PY

W

U V(3x)o

(8x)o

(8x)o

(5x)o

o100o

60o60o

o

o

QP

X

U V

W

60

150

E

B

C

G

AFH

P

A

D

C

B2x+35

3x

2x

x+45

C

B

x)

(x)

o

o

B

A

C

(x)o

(2x+30)o

P P (x)

60

Y

M

N

X

PY

W

U V(3x)o

(8x)o

(8x)o

(5x)o

o

o

100o

60o60o

o

o

x x x x

x

x

mAD mCB x

mDB

+ + + ===

= = =

5 5 360

12 360

30

300

== =5 150x o

x x

x

x

x

mAB x

mBC

o

+ + =+ =

==

= =

2 30 180

3 30 180

3 150

50

50!

= + = ( ) + =2 30 2 50 30 130x o

3 8 8 5 360

24 360

15

3 3 15 45

x x x x

x

x

mUV x o

+ + + ===

= = ( ) =

mmVW mWY x

mYU x

o

o

= = = ( ) =

= = ( ) =

8 8 15 120

5 5 15 75

x

mNX mMY x

mNY mMX x

o

o

=

= = =

= = =

60

60

180 120

!

-


mAD = mCB = x = 30°

( (

mDB = 5x = 150°

(

mAB = x = 50°

(

mBC = 2x + 30 = 2(50) + 30 = 130°

(

mUV

(

mVW

(

mWY

(

mYU

(

mNX

(

mMY

(

mNY

(

mMX

(

NameUnit II, Part C, Lesson 1, Quiz Form A—Continued—


4. State the missing reason (mathematical property) for each step in the following proof.

Step 1 __________________________________

Step 2 __________________________________

Step 3 __________________________________

Step 4 __________________________________

5. Which of the following statements are true for all real numbers a and b? If false, give a numerical example to show it is false.

a) If a < b, then a + b < 0. b) If a < b, then a • b > 0.

3

4=

1

3⋅ x Assumed to be true

Multiplication Property of Equality

Associative Property of Multiplication

Property of Multiplicative Inverse

Property of One for Multiplication

4

3

3

4=

4

3

1

3⋅ ⋅

⋅x

4

3

3

4=

4

3

1

3⋅

⋅ ⋅x

1 =4

9⋅ x

x =4

9

False

a < b a + b < 03 < 5 3 + 5 < 0

8 < 0 (not true)

False

a < b a • b > 0-3 < 4 (-3) • (4) > 0

-12 > 0 (not true)


Name

Class Date Score

Quiz Form B

Unit II - Fundamental TermsPart C - Postulates (or Axioms)Lesson 1 - Need

1. Name the property of equality (reflexive, symmetric, or transitive) that allows you to conclude that eachstatement is true.

a) MN = MN ___________________________

b) If x = 3, then 3 = x ___________________________

c) If m/R = m/S and m/S = m/T,

then m/R = m/T ___________________________

2. Write a paragraph proof to show that the following is true:

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

3. Use the given property in each of the following to draw a conclusion.

a) Transitive Property: If a = b and b = c, then a = c

m/A = m/C, m/C = 61

Conclusion:_____________________________________

b) Symmetric Property: If m = n, then n = m.

AB = CD

Conclusion:_____________________________________

c) Multiplication Property: If a = b and c = d, then a • c = b • d

Conclusion:_____________________________________

Reflexive Property of Equality

Symmetric Property of Equality

Transitive Property of Equality

Since , it follows that by the addition property for equality, (add 9 to both sides of equation). Multiplying

both sides by 5, the multiplication property for equality leads to x = 125.

m/A = 61

CD = AB

x + 7 = 13

If5

- 9 =16,� then� = 125x

x

4 + 7 = 52, � �1

4= .25x( )

x

5- 9 = 16

x

5= 25

NameUnit II, Part C, Lesson 1, Quiz Form B—Continued—


4. State the missing reason (mathematical property) for each step in the following proof.

Step 1 __________________________________

Step 2 __________________________________

Step 3 __________________________________

Step 4 __________________________________

5. Which of the following statements are true for all real numbers a and b? If false, give a numerical example to show it is false.

a) If a < b, then a – b > 0. b) If a < b, then a 4 b > 0. (b ? 0)

−2 < 8x Assumed to be true

Negative Multiplication Property for Inequality

Associative Property of Multiplication

Property of Multiplicative Inverse

Property of One for Multiplication

1

22 >

1

28−

−−( ) ⋅x

1

22 >

1

28−

−−⋅

⋅ ⋅x

1 > 4⋅ −x

x > − 4

False

a b<

<−3 5

a b b

not

÷ > ≠

÷ >

>

−

−

0 0

3 5 0

3

50

)

true)

(

(

False

a < b a – b < 03 < 5 3 – 5 < 0

-2 < 0 (not true)


Name

Class Date Score

Quiz Form A

Unit II - Fundamental TermsPart C - Postulates (or Axioms)Lesson 2 - Postulate 1(Existence of Points)Lesson 3 - Postulate 2(Uniqueness of Lines, Planes and Spaces)Lesson 4 - Postulate 3(One, Two, and Three Dimensions)

In the given figure, points X, Y, P and Z lie in the same plane with point W not in plane XYPZ. Give a reason for each statement in problems 1- 5.

1. Exactly one plane contains W, X, and Y.

________________________________________________________

________________________________________________________

2. Points Z and P determine a line.

________________________________________________________

________________________________________________________

3. Any plane containing X and Y must contain XY.

________________________________________________________

________________________________________________________

4. Point X is not on YZ.

________________________________________________________

________________________________________________________

5. Point X is not on plane WYZ.

________________________________________________________

________________________________________________________

PC

mk

EF

D

B

n

l

A B C D W

P Z

X Y

X

H G

BA

F CE

D A

H G

BA

F CE

D

Q

B

A

If there is a plane, then there exist at least 3 different, non-collinear points on the plane.

If we have 2 different points, then we have exactly one line containing those two points.

If those points are in a plane, then the line containing them must be in the plane.

For any line in a plane, there is at least one point in the plane that is not on the line.

For every plane in space, there is at least one point in space that is not on the plane.


Name

Class Date Score

Quiz Form B

Unit II - Fundamental TermsPart C - Postulates (or Axioms)Lesson 2 - Postulate 1(Existence of Points)Lesson 3 - Postulate 2(Uniqueness of Lines, Planes and Spaces)Lesson 4 - Postulate 3(One, Two, and Three Dimensions)

In the given figure, Plane M contains point D, and collinearpoints A, B, and C. Point E is not in Plane M. Give a reason for each statement in problems 1 - 5.

1. There is exactly one plane containing points E, B, and A.

________________________________________________________

________________________________________________________

2. There is exactly one line containing points D and A.

________________________________________________________

________________________________________________________

3. Line DC is in Plane M.

________________________________________________________

________________________________________________________

4. Point D is not on AC.

________________________________________________________

________________________________________________________

5. Point D is not on plane ACE.

________________________________________________________

________________________________________________________

P

mk

EF

D

B

n

B C D W

P Z

X Y

D

C

E

A

BX

Y

D

B

C

A

E

M

H G

BA

F CE

D

llA

B

M

k

P

RS

R

H G

BA

F CE

Dm k

A

BC

Q

M

Nll

A

B

A

C

A B C

012345678

P TQ R S

B A C

P

A B

There is exactly one plane through three different, non-collinear points.

There is exactly one line through two different points.

For any two different points in a plane, the line containing them is in the plane.

For any line in a plane, there is at least one point in the plane that is not on the line.

For every plane in space, there is at least one point in space that is not on the plane.

NameUnit II, Part C, Lessons 2, 3& 4, Quiz Form B—Continued—


In the given figure, plane M intersects plane N. Use this information to answer problems 6-14.

6. Name two points that determine line ,. ____________

7. Name three points that determine plane M. ____________

8. Name the intersection of plane M and plane N. ____________

9. Does AD lie on plane M?. ____________

10. Does plane N contain any points not on AB?. ____________

11. Only one ____________ contains AB and point D.

12. The plane containing points A, B, and C also contains point ____________.

13. Line CE and line AB intersect at point ____________.

14. Line CE and line AB determine plane ____________.

In problems 15-18, tell how many planes at most can pass through each set of points.

15. One line ____________

16. Two points ____________

17. Three collinear points ____________

18. A line and a point not on the line ____________

W

P Z

X Y

D

C

E

A

BX

Y

D

B

C

A

E

M E

A B

DN

MC

H G

BA

F CE

D

llA

B

M

k

P

RS

Xk

P

RS

H G

BA

F CE

Dm k

A

BC

Q

M

Nll

AB

B

A

C

A B C

0 1 2 3 4 5 6 7 812345678

P TQ R S

B A C

P

N

P TQ R S

A B

A

A

O C

B

A B C D E F

ll

C D

C and E

C, E, and B

AB

Yes

Yes

plane

D

A

M

infinite

infinite

infinite

one

Name

Class Date Score

Quiz Form A

Unit II - Fundamental TermsPart C - Postulates (or Axioms)Lesson 4 - Postulate 3 (One, Two, and Three Dimensions)Lesson 5 - Postulate 4 (Separation of Lines, Planes and Spaces)Lesson 6 - Postulate 5 (Intersection of Lines or Planes)


1. Planes M and N are known to intersect. Describe the intersection of planes M and N? ________

2. State the Postulate that supports your answer to problem 1. ___________________________________

___________________________________________________________________________________

3. Sketch a diagram to illustrate the geometric relationships in problems 1 and 2.

4. Is it possible for three non-collinear points to all lie in each of two planes? Explain.

___________________________________________________________________________________

___________________________________________________________________________________

5. Is it possible for two points to lie in plane X, two other points lie in a different plane Y, and for all four points to be coplanar but not collinear? Explain.

___________________________________________________________________________________

___________________________________________________________________________________

6. What is the greatest number of planes determined by four points, no three of which are collinear? _______

7. Two distinct lines intersect in at most __________________.

8. A plane is determined by three ___________________ points.

(1 point or 2 points)

(collinear or non-collinear)

A line

If two different planes intersect, then the intersection

is a unique line.

P

W

P Z

X Y

D

C

E

A

BX

Y

D

B

C

A

E

M E

A B

DN

MC

H G

BA

F CE

D

llA

B

M

k

P

RS

Xk

P

RS

H G

BA

F CE

Dm k

A

BC

Q

M

Nll

AB

B

A

C

A B C

0 1 2 3 4 5 6 7 812345678

P TQ R S

B A C

P

QM O

N

P TQ R S

A B

I

JG O

H

A

A

A

O

O

O

C

B

B

B

A B C D E F

ll

C D

No; The planes would have to be the same plane. From II-C-3, there is exactly one plane through three different, non-collinear points.

Yes; The two points in Plane X determine a line. The two points in Plane Y determine a line. If the two lines are parallel, the 4 points

will be coplanar.

4 (see below)

1 point

non-collinear

PC

mk

EF

D

B

n

l

A B C D


NameUnit II, Part C, Lessons 4,5&6, Quiz Form A—Continued—

To answer problems 9-12, you will have to visualize certain linesand planes not drawn specifically in the diagram.

9. Name a plane that contains AC. ____________

10. Name the intersection of plane DCFE and plane ABCD. ____________

11. Name two lines that are not shown in the diagram and that do not intersect. ____________

12. Name three planes that do not intersect EF. __________________________

In the figure at the right, line , separates plane M into two half-planes.Use this information to answer problem 13.

13. Point A and point B are in different half-planes. Justify this statement by using the appropriate part of the line separation postulate.

______________________________________________

______________________________________________

______________________________________________

______________________________________________

PC

mk

EF

D

B

l

P Z

X Y

X

H G

BA

F CE

D A

H G

BA

F CE

Dm

Q

B

A

P

QM O

N

P Q R S

A

plane ABC

DC

AH, BGAnswers may vary

plane DHG, plane ABC, plane ABG

PC

mk

EF

D

B

n

l

A B C D W

P Z

X Y

D

C

E

A

BX

Y

D

A

M

H G

BA

F CE

D

llA

B

M

P

R

H G

BA

F CE

Dm k

A

BC

Q

M

ll

B

A

C

A B C

078

P Q R S

B

P TQ R S

A

C

The line segment joining point A to point B will intersect separation

line in one point.


Name

Class Date Score

Quiz Form A

Unit II - Fundamental TermsPart C - Postulates (or Axioms)Lesson 7 - Postulate 6 (Ruler)Lesson 8 - Postulate 7 (Protractor)

1. Complete the following relations:

a) _______ c) _______ e) _______

b) _______ d) _______ f) _______

2. Using the given number line, find the distance between each indicated pair of points.

a) B and A b) A and C c) F and D d) C and F

e) C and D f) E and A g) D and B h) A and F

3. How many different points can be either to the right or to the left of a given point on a horizontalnumber line? ________________________________________________________________________

___________________________________________________________________________________

7 = 0 = -19 - -3 =

-7 = 6 - 9 = 9 -6⋅ ( ) =

DD C

M E

DN

llA

B

M

k

P

RS

Xk

P

RS

m k

A

BC

Q

M

Nll

AB

A

C

A B C

0 1 2 3 4 5 6 7 812345678

P TQ R S

B A C

P

Q

TR S

B

I

JG O

H

A

A

A

O

O

O

C

C

C

B

B

B

A B C D E F

ll

C D

7 0 16

7 3 54

− − −( )=

4 6

2 2

0 6

6 6

− −( )=

8 3

5 5

−=

0 8

8 8

−− =

0 3

3 3

−− =

5 6

11 11

− −( )=

3 4

7 7

− −( )=

− −− =

6 8

14 14

An infinite number to the right; an infinite number to the left.

NameUnit II, Part C, Lessons 7& 8, Quiz Form A—Continued—


4. Suppose on line t, point P corresponds to the real number 2. How many different points exist on t such that the distance from P to each of these points is 12? __________What real numbers correspond to these points? __________ How many different points exist on t such that the distance from P to each of these points is a given real number q? __________What are the coordinates of these points?__________

5. If AB = 6 and AC = 38, is BC necessarily 32? Explain your answer. _______________________________

______________________________________________________________________________________

______________________________________________________________________________________

6. In the given diagram of collinear points, PT = 20, QS = 6, and PQ = QR = RS. Find each length asked for below.

a) QR b) ST c) RT d) SP

7. Measure the given line segment to the nearest millimeter.

CD = _______

8. Use a protractor to determine the measure of each angle named below.

a) m/NOP = _________________

b) m/NOM = _________________

c) m/QON = _________________

G

B

F C

llA

B

M

k

P

RS

Xk

P

RS

G

B

F C

m k

A

BC

Q

M

Nll

AB

B

A

C

A B C

0 1 2 3 4 5 6 7 812345678

P TQ R S

B A C

P

QO

P TQ R S

A B

I

JG O

H

A

A

A

O

O

O

C

C

C

B

B

B

A B C D E F

l

C D

PC

EF

D

B

l

P Z

X Y

D

CA

BX

H G

BA

F CE

D

llA

B

M

H G

BA

F CE

Dm k

A

BC

Q

B

A

C

A B C

0

P

QM O

N

P Q R S

A

G

H

D

CA

BD

B

C

A

M E

A B

DN

ll

B

M

k

P

RS

Xk

P

RS

m k

A

BC

M

Nll

AB

A

C

B C

0 1 2 3 4 5 6 7 812345678

P TQ R S

B A C

P

TR S

B

I

JG O

H

A

A

A

O

O

O

C

C

C

B

B

B

A B C D E F

ll

C D

2

2

No; BC is 32 if B is between A and C.

BC is 44 if A is between B and C.

QR + RS = QS

QR - RS = QS = 6

RS + RS = 6

2 RS = 6

RS = 3

⋅

20 - 3(3)

ST = 11

RS + ST = RT

3+11 = RT

14 = RT

SP = PQ +QR + RS

SP = 3+ 3+ 3

SP = 9

54 mm

90O

72O

108O

14; -10p + q; p - q

2 ; 2 +12 = 14 ; 2 ; p + q

2 - 112 = -10 p - q

QR + RS

NameUnit II, Part C, Lessons 7& 8, Quiz Form A—Continued—


9. If M is between L and N and , find x, LM, MN and LN by

using the Segment Addition Postulate.

10. Explain why the Angle Addition Property begins with “If a ray OB lies between rays OA and OC,...” Use a sketch of /AOB with ray OB not between rays OA and OC to illustrate what the relationship

would be otherwise. _____________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

LM =1

2+2,� MN = 3 +

3

2and� LN = 5 +2x x x,

LM + MN = LN

1

2x + 2 + 3x +

3

2= 5x + 2

31

2x + 3

1

22= 5x + 2

11

2= 1

1

2x

1 = x

LM =1

21 + 2 = 2

1

2

MN = 3 1 +3

2= 4

1

2LN = 5 1 + 2 = 7

( )

( )( )

If OB was in the exterior of /AOC (in other words, not between OA and OC), then

either /BOA or /BOC would be greater than /AOC and you couldn't add the measures properly.

M

kRXkR

M

Nll

AB

0 1 2 3 4 5 6 7 812345678

P TQ R S

B A C

P

T

B

I

JO

A

A

A

O

O

O

C

C

C

B

B

B

A B C D E F

l

C D

m k

A

BC

Q

M

Nll

AB

B

A

C

A B C

0 1 2 3 4 5 6 7 812345678

P TQ R S

B A C

P

Q

P TQ R S

A B

I

JG O

H

A

A

A

O

O

O

C

C

C

B

B

B

A B C D E F

l

C D


Name

Class Date Score

Quiz Form B

Unit II - Fundamental TermsPart C - Postulates (or Axioms)Lesson 7 - Postulate 6 (Ruler)Lesson 8 - Postulate 7 (Protractor)

1. Complete the following:

a) _______ c) _______ e) _______

b) _______ d) _______ f) _______

2. Find the distance between each pair of points having the given coordinates.

a) —3 and 10 b) —2.4 and —3.1 c) d) 3.14 and 7.3

e) f) g) 11.3 and 7.1 h) a and b

3. Suppose line t contains M and Q. Does a point P always exist such that Q is between M and P? ______ Does a point R always exist such that M is between R and Q? __________What in general does this tell us? _____________________________________________________________________________

___________________________________________________________________________________

___________________________________________________________________________________

0 9− = 3

4= − =0 76.

- 5 3+ = −( ) −( ) =5 3 − =13

71

8and� - 3

1

4

5

8and�

1

27 and� 11

9 .76

13158

3

4

10 3

13 13

− −( )=

− − −( )− =

3 1 2 4

0 7 0 7

. .

. .− −

− −

− + −

− =

31

47

1

8

13

4

57

8

26 57

8

83

8

83

810

3

8or

7 3 3 14

4 16 4 16

. .

. .

−=

1

2

5

8

4

8

5

8

1

8

1

8

−

−

− =

11 7

3 32 2 65

67 67

−

−=

. . .

. . .

approx

approx

7 1 11 3

4 2 4 2

. .

. .

−− =

b a or a b− −

yes

Between any two given points there always exists another point.

yes


NameUnit II, Part C, Lessons 7& 8, Quiz Form B—Continued—

4. How many different points on a line can be half the distance from one given point to another given point? Explain your answer. ___________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

5. If A, B, and C are collinear, AB = 26, and BC = 16, is AC necessarily 42? _________Explain your answer.

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

6. In the accomplanying diagram of collinear points, PT = 26, PR = 8, and PQ = QR = RS. Find each length asked for.

a) RS b) PQ c) QS d) QT

7. Measure the given line segment to the nearest millimeter.

AB = _______

8. Use a protractor to determine the measure of each angle named below.

a) m/GOI = _________________

b) m/HOJ = _________________

c) m/IOH = _________________

BAM

H G

BA

F CE

Dm k

A

BC

Q

M

Nll

AB

B

A

C

A B C

0 1 2 3 4 5 6 7 812345678

P TQ R S

B A C

P

QM O

N

P TQ R S

A B

I

JG O

H

O

O

B

B

A B C D

C D

P

E

D

B

X Y

D

BX D CM

H G

BA

F CE

D

llA

B

M

k

P

RS

R

H G

BA

F CE

Dm k

A

BC

Q

M

Nll

A

B

A

C

A B C

012345678

P TQ R S

B A C

P

QM O

N

P TQ R S

A B

I

JG O

H

A B C

C D

Y

DM E

DN

G

B

F C

llA

B

M

k

P

RS

Xk

P

RS

G

B

F C

m k

A

BC

Q

M

Nll

AB

B

A

C

A B C

0 1 2 3 4 5 6 7 812345678

P TQ R S

B A C

P

Q

P TQ R S

A B

I

JG O

H

A

A

A

O

O

O

C

C

C

B

B

B

A B C D E F

ll

C D

One; The distance between any two points divided by 2 is unique.

AC is 42 only if B is between A and C. AC is 10 if C is between A and B.

PQ QR PR

PQ QR

QR QR

QR

QR

RS

+ ==

+ =⋅ =

==

8

2 8

4

4

PQ QR RS

PQ

PQ

= == ==

4 4

4

QR RS QS

QS

QS

+ =+ =

=4 4

8

PT PQ QT

QT

QT

− =− =

=26 4

22

82 mm

135O

146O

103O

No

NameUnit II, Part C, Lessons 7& 8, Quiz Form B—Continued—


9. If M is between L and N and , find x, LM, MN and LN

by using the Segment Addition Postulate.

10. What requirement for adding segment lengths is similar to the requirement that a ray is between two rays when adding angles? ___________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

LM =1

3+ 4,� MN = 4 +

4

3, and� LN = 4 + 6

1

3x x x

LM MN LN

x x x

x

+ =

+

+ +

= +

+

1

34 4

4

34 6

1

3

41

355

1

34 6

1

31

31

3

= +

=

=

x

x

x

LM

MN

LN

= ( ) + =

= ( ) + =

= ( ) + =

1

33 4 5

4 34

313

1

3

4 3 61

318

1

3

In the Segment Addition Postulate, the requirement that a point must be between two points, is

similar to the requirement in the Angle Addition Postulate that a ray must be between two rays.


NameUnit II, Part C, Lessons 9, 10 &11, Quiz Form A—Continued—

Using the information on the circle at the right, find the distance between each pair of points in problems 3-7.Give two distance measurements in each case.

3. B to H 4. G to C

Minor Arc Major Arc Minor Arc Major Arc

5. D to A 6. F to D 7. C to B

Minor Arc Major Arc Minor Arc Major Arc Minor Arc Major Arc

8. How many lines can be perpendicular to a given plane from a given line not in the given plane?_______________________________________________________________________________________________

9. How many lines can be perpendicular to a given line at a given point on the given line?________________

______________________________________________________________________________________

Mll

B

RS 0

90

180

270

A

C

G

E

335198

77132

BD

FH

0

90

180

270

A

C

G

E

290253

32

119B

D

F H

Mll

B

RS

Mll

B

B

RS

MllR

S

Mll

B

RS

Mll

B

RS

360-335( ) +

+

77

25 77

102

102o

360 102

258

−o

270 90

180

180

−

o

360 180

180

−o

132 0

132

132

−

o

360 132

228

−o

198 132

66

66

−

o

360 66

294

−o

90 77

13

13

−

o

360 13

347

−o

infinite

infinite

NameUnit II, Part C, Lessons 9, 10 &11, Quiz Form B—Continued—

64 © 2014 VideoTextInteractive Geometry: A Complete Course

Using the given circle, find the distance between the following pairs of points. Give two distance measurements in each case.

3. D to E 4. C to H

Minor Arc Major Arc Minor Arc Major Arc

5. F to G 6. B to E 7. D to G

Minor Arc Major Arc Minor Arc Major Arc Minor Arc Major Arc

In the given figure, line , passes through points R and S. point B is not on line ,. points R, S, and B are in plane M.Use this information to solve problems 8 and 9 below.

8. State the postulate that guarantees the existence of only one line through point B, which is perpendicular to line ,.

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

9. Trace the figure in Exercise 8, and complete the picture by sketching the perpendicular line through point B to line ,.

270

A

0

90

180

270

A

C

G

E

290253

32

119B

D

F H

Mll

B

RS

Mll

B

B

RS

MllR

S

Mll

B

RS

Mll

B

RS

119 180

61

61

−−

o

360 61

299

−o

360 290 90

70 90

160

160

−( ) +

+

o

360 160

200

−o

270 253

17

17 0

− 360 17

343

−o

180 32

148

148

−

o

360 148

212

−o

270 119

151

1510

− 360 151

209

−o

M F

180

Mll

B

RS

Mll

B

B

RS

MllR

S

Mll

B

RS

Mll

B

RS

Mll

B

RS 0

90

180

270

A

C

G

E

335198

77132

BD

FH

0

90

180

270

A

C

G

E

290253

32

119B

D

F H

Mll

B

RS

Mll

B

B

RS

MllR

S

Mll

B

RS

Mll

B

RS

If you have a line in a plane, and a point in the plane, not on the given line, then there is one and only one line through

the point, which is perpendicular to the given line.

NameUnit II, Fundamental Terms, Unit Test Form A—Continued, Page 2—


_______13. If there is space...

_______14. placeholder symbols

_______15. supplementary angles

_______16. If on a line point B lies between points A and C, then AB + BC = AC. This is called the...

_______17. arc

_______18. line

_______19. half plane

_______20. Property of Zero for Addition

_______21. angle bisector

_______22. definition

_______23. radius

_______24. The union of two distinct rayswith a common endpoint is an...

_______25. acute angle

_______26. If, in a half plane, a ray OB liesbetween rays OA and OC, then...

(C-2)

(A-1)

(B-8)

(C-7)

(A-2)

(B-6)

(C-1)

(B-8)

(B-1)

(B-9)

(C-8)

(B-7)

(C-8)

(B-9)

n) The union of the separation point of a line with a half-line.

o) A description of a term which clearly specifies what the term means.

p) Distance from the center of the circle to the curve which makes the circle.

q) There exists a unique real number 0, such that for any real number a, a + 0 = a.

r) An angle smaller than a right angle.

s) containing the two points is in the plane.

t) If a = b and b = c, then a = c

u) of a complete revolution.

v) If a = b, then a + c = b + c

w) angle

x) Segment Addition Property

y) m/AOB + m/BOC = m/AOC

z) One of the two sets of points on a plane on either side of a specific line called the separation line.

1

360

c

b

g

x

i

a

z

q

f

o

p

w

r

y

Unit II, Fundamental Terms, Unit Test Form A—Continued, Page 3—


In problems 27-30, sketch and label the figures described.

27. Points A, B, C, and D are coplanar, but 28. Line ,, on plane M, point P on line ,, line kA, B, and C are the only three of those intersects plane M through point P.points that are collinear.

29. Plane M contains intersecting 30. Plane X and Y intersect in AB.lines j and k.

Name

1 5 x2

N O P Q

EB C

D

87O

OG

F

ED

C

B

AH

X A

B Y

P C

(3x)

(3x-3)

(2x+15)

B

A

M

l

A T SQ

RX

Y

AB

C

D

Pl

k

M M k

j

Y

A

B 45 144O O

60O

120O B C

A

B C

A

O

O

O

1 5 x25

M N O P Q

EA B C

D

87O

OG

F

ED

C

B

AH

X A

B Y

P C

(3x)

(3x-3)

(2x+15)

B

A

M

l

A T SQ

RXW

YZ

AB

C

D

Pl

k

M M

Y

XA

B 45 144O O

60

120O B C

A

B C

A

O

O

O

A

Y

C(3x-3)

(2x+15)

B

A

M

l

Pl

k

M M k

j

144O O

60O

O

O

1 5 x25

M N O P Q

EA B C

D

87O

OG

F

ED

C

B

AH

A T SQ

RXW

YZ

AB

D

Y

XA

B 45 1O

120O

A

(B-2) (C-6)

(C-4) (C-6)



For problems 31-34, use the figure below.

31. Name a point in NP that is not on NP. ________________

32. Complete: MN = ____________ and NO = ____________.

33. Complete: MN and NO are called ________________________ segments.

34. If point P is the midpoint of OQ, find the value of x.

1 5 x25

M N O P Q

EA B C

D

87O

OG

F

ED

C

B

AH

X A

B Y

P C

(3x)

(3x-3)

(2x+15)

BM

A T SQ

RXW

YZ

AB

C

D

Pl

k

M M

Y

XA

B 45 144O O

120O B C

A

B C

A

O

O

Opoint Q

3 3

-2 - 5(-5)

-2 + 5

3

3

1- (-2)

1+ 2

3

3

congruent

5 -1

4

4

5 + 4 = 9

P

(B-4)

(B-5)

(B-5)

(B-5)

Point Q has 9 for a coordinate



For problems 35-43, use the figure below.

35. Name three angles that have vertex D. _______________________

36. Name two pairs of angles that form a linear pair. ________________________________.

37. Name a pair of adjacent angles that do not form a linear pair. ________________________

38. m/CBD = ____________.

39. What kind of angle is /EAD? ________________________

40. m/ABD + m/DBC = ____________.

41. Assume DB bisects AC, AB = 5x – 3 and BC = x + 25. Find the value of x. ____________

42. /DBC is an ________________________ angle. (classify by referring to types of angles)

43. /ABD is an ________________________ angle. (classify by referring to types of angles)

1 5 x25

M N O P Q

EA B C

D

87O

OG

F

ED

C

B

AH

X A

B Y

P C

(3x)

(3x-3)

(2x+15)

B

A

M

A T SQ

RXW

YZ

AB

C

D

Pl

k

M M

Y

XA

B 45 144O O

120O B C

A

B C

A

O

O

O

(B-6)

(B-8)

(B-8)

(B-7)

(C-7)

(B-5)

/ADB, /BDC, /ADC

/EAD and /DAC or /ABD and /DBC

/ADB and /CDB

93O

obtuse

180O

7

obtuse

acute

5x - 3 = x + 25

4x = 28

x = 7



In problems 44-47, classify each statement as true or false.

44. It is possible for two intersecting lines to be non-coplanar. ____________

45. It is possible to locate three points in such a position thatan unlimited number of planes contain all three points. ____________

46. Through any three points there is at least one line. ____________

47. If points A and B lie in plane P, then so does any pointof AB. ____________

For problems 48-51, use a protractor to draw angles having the following degree measures

48. 45O 49. 60O

50. 144O 51. 120O

1 5 x

O P Q

EC

OG

F

ED

C

B

AH

X A

B Y

P C

(3x)

(3x-3)

(2x+15)

B

A

M

l

S

RA

B

C

D

Pl

k

M M k

j

45 144O O

60O

B C

A

B C

A

O

O

O

5 x

P Q

E

O

ED

C

B

AH

X A

B Y

P C

(3x)

(3x-3)

(2x+15)

B

A

M

l

B

C

D

Pl

k

M M k

j

45 144O O

60O

B C

A

B C

A

O

O

O

x

Q

X A

B Y

P C

(3x)

(3x-3)

(2x+15)

B

A

M

l

CP

lk

M M k

j

144O O

60O

A

O

O

O

1 5 x25

M N O P Q

EA B C

D

87O

OG

F

ED

C

B

AH

X

B

A T SQ

RXW

YZ

AB

C

D

Y

XA

B 45O

120O B C

A

(C-4)

(C-4)

(C-4)

(C-4)

(C-8)

(C-8) (C-8)

(C-8)

false

true

false

true



In the figure at the right, the following pairs of angles are complementary: /AOB and /BOC, /COD and/DOE, /EOF and /FOG, /GOH and /HOA. If m/AOB = 45O, m/COD = 30O, /EOF > /FOG, andm/HOA = 10O, use this information to find the measures asked for in problems 52-56.

52. m/BOC ____________

53. m/DOE ____________

54. m/GOH ____________

55. m/GOF ____________

56. m/FOE ____________

In the figure at the right, XY and AB are diameters.Use the information given in the figure to solve problems 57-66.

57. Find the value of x.

58. Find m/BPY ____________ 59. Find mYAX ____________

60. Find m/YPC ____________ 61. Find m/BPC ____________

62. Find mBX ____________ 63. Find m/CPA ____________

64. Find m/XPA ____________ 65. Find mCA ____________

66. Find mBC ____________

1 5 x25

M N O P Q

EA B C

D

87O

OG

F

ED

C

B

AH

X

B

A T SQ

RXW

YZ

AB

C

D

Y

XA

B 45O

120O B C

A

((

((

3x + 3x - 3+ 2x +15 = 180

8x +12 = 180

8x = 168

x = 21

1 5 x25

M N O P Q

EA B C

D

87O

OG

F

ED

C

B

AH

X A

B Y

P C

(3x)

(3x-3)

(2x+15)

B

A

M

A T SQ

RXW

YZ

AB

C

D

Pl

k

M M

Y

XA

B 45 144O O

O

O

O

(C-8)

(C-8)

(C-8)

(C-8)

(C-8)

45O

60O

80O

45O

45O

(C-9)

63 O

60 O

117O

63O

123O

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

180O

123O

57O

57O

Name


Unit II, Fundamental Terms, Unit Test Form B—Continued, Page 2—

_______13. closed phrase

_______14. If there is a plane,...

_______15. edge

_______16. plane

_______17. minor arc

_______18. equal angles

_______19. betweeness (points)

_______20. obtuse angle

_______21. open sentence

_______22. bisector of a line segment

_______23. chord

_______24. Symmetric Property of Equality

_______25. Property of One for Multiplication

_______26. If you have a line in a planeand a point in the plane noton the given line, then...

(A-1)

(C-2)

(B-6)

(A-2)

(B-8)

(B-2)

(B-7)

(A-1)

(B-5)

(B-9)

(C-1)

(C-1)

(C-11)

(B-9)

m) Symbols that represent operations on numbers, or, more generally, symbols which indicate the actions of mathematics.

n) If a = b, then b = a

o) Any point, line segment, ray, line, or plane which intersects the line segment in the midpoint of the line segment.

p) Exactly one space containing those four points.

q) Geometric tool used to measure the number of degrees in an angle.

r) If a = b, then a • c = b • c

s) Line segment across a circle whose endpoints lie on the curve forming the circle.

t) There exists a unique real number 1, such that for every real number a, a • 1 = a.

u) Mathematical expression which contains neither a relation symbol, nor a placeholder symbol.

v) (a + b) + c = a + (b + c)

w) that is not on the line.

x) The line in a plane which separates the plane into two half planes.

y) Two angles with the same measure.

z) A geometric figure which is described as an infinite set of points, arranged as a flat surface, extending to infinity in both directions.

u

h

x

z

g

y

d

k

b

o

s

n

t

j

Name


Unit II, Fundamental Terms, Unit Test Form B—Continued, Page 4—

Answer problems 32-43, using the given figure..

32. Name four collinear points. ____________________

33. Name the intersection of DG and BC. ____________________

34. What property justifies the statement AF + FC = AC. ________________________________

35. If BD bisects AC, name two congruent segments. _______________________

36. Name a right angle. ________________________________

37. Name three coplanar points. ________________________

38. Is it possible for an answer to exercise 37 to be incorrect? Explain. _______________________________

39. Name a pair of complementary angles. ________________________

40. Name a pair of supplementary angles. ______________________

41. Name two pairs of vertical angles. _____________________________________

42. If m/AFB is 28O, find m/EFC _________________ and m/AFE. _________________

43. If m/ACB is 12O, find m/ACE. ________________________

Q

P

G

E

CB

AF D

P

TV

Q

R

S

T

S

U

V

Q

R

900

360

260

780

1300

(B-2)

(B-3)

(C-1)

(B-5)

(B-7)

(B-2)

(B-2)

(B-8)

(B-8)

(B-8)

(B-8)

(B-8)

A, F, G, C

Point B

Segment Addition Property

AG > GC

/BCE

A, B, C (answers will vary)

No, any combination of points

in the figure will be coplanar.

/ACB and /ACE

/AGB and /AGD

/AGB and /DGC; /AGD and /BGC

28O

78O

152O

NameUnit II, Fundamental Terms, Unit Test Form B—Continued, Page 5—


For exercises 44-48, refer to a numberline, not pictured here, with the following information. Point A hascoordinate 2 and point B has coordinate 5.

44. What is the length of AB. ____________________

45. What is the coordinate of the midpoint of AB? ____________________

46. If A is the midpoint of PB, what is the coordinate of point P? ______________________

47. What is the coordinate of a point that is on AB and is 4 units from B. _______________________

48. What is the coordinate of a point that is 4 units from point B, but is not on AB? _____________________

49. Is it possible for a line and a point to be non-coplanar? ________________________

50. Is it possible for the intersection of two planes to consist of a segment? ____________________________

In the figure at the right, VR bisects /QVS, m/PVQ = 72O, m/TVS = 70O, and VP and VT are opposite rays.Use this information to find the measures asked for in problems 51-56.

51. m/QVS ____________

52. m/QVR ____________

53. m/PVR ____________

54. m/TVR ____________

55. m/PVS ____________

56. m/TVQ ____________

J Q

PM

N

HG

E

CB

AF D

P

TV

Q

R

SU

780

1300

A B2 5

(B-5)

(B-5)

(B-5)

(B-5)

(B-5)

(B-2)

(C-5)

(C-8)

(C-8)

(C-8)

(C-8)

(C-8)

(C-8)

J Q

PM

N

HG

E

CB

AF D

A B2 5

5 2- or 3

23

23

1

2+

or

—1 (3 to the left of 2)

(5 + 4) or 9

1 (4 to the left of 5)

no

no

38O

19O

91O

89O

110O

108O



Complete each of the statements in problems 57-62.

57. m/A = m/A is true because of the ____________________ Property of Equality.

58. If m/A = m/A, then is it true that /A > /A?________________________________

59. If m/B = m/C and m/C = m/A, then m/B = m/A. This is an example of the

____________________ Property of Equality.

60. With reference to exercise 59, would it be correct to say that if /B > /C and /C > /A,

the /C > /A? ____________________

61. If m/A = m/B, then m/B = m/A is an example of the ____________________ Property.

62. If /A > /B, then /B > /A must be true since if two angles have the same measure (in exercise 61)

then by definition, the angles are ____________________.

Tell if each of the statements in problems 63-65 is true or false..

63. Angle congruence is reflexive. _______________

64. Angle congruence is transitive. _______________

65. Angle congruence is not symmetric. _______________

(C-1)

(B-8)

(C-1)

(B-8)

(C-1)

(B-8)

(C-1)

(C-1)

(C-1)

Reflexive

yes (definition of congruence)

Transitive

yes

Symmetric

congruent

true

true

false



In the figure at the right, the measure of each angle in circle Q is given. Use this information to find themeasure of each arc in problems 66-73.

66. RT ____________

67. UR ____________

68. VS ____________

69. US ____________

70. RSU ____________

71. SUV ____________

72. TVR ____________

73. VTR ____________

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

Q

P

G

E

CB

AF D

P

TV

Q

R

S

T

S

U

V

Q

R

900

360

260

780

1300

((

((

((

((

126O

104O

116O

166O

256O

244O

234O

334O

Name


Unit II, Fundamental Terms, Unit Test Form C—Continued, Page 2—

_______13. If you have a line in a planeand a point in the plane noton the given line, then...

_______14. a > b means...

_______15. half space

_______16. Commutative Propertyof Multiplication

_______17. If, on a circle, a point B liesbetween point A and C, then...

_______18. plane

_______19. Additive Inverse Property

_______20. complementary angles

_______21. separation point

_______22. angle

_______23. Transitive Property of Order

_______24. If there is a line,...

_______25. For an two different points in a plane,the line containing these points...

_______26. Reflexive Property of Equality

(C-10)

(C-1)

(B-6)

(C-1)

(A-2)

(C-1)

(B-8)

(B-4)

(B-6)

(C-1)

(C-2)

(C-4)

(C-1)

(C-9)

n) Represented by a straight line segment with arrowheads on each end.

o) Two angles whose measures total 90 degrees.

p) If a < b and b < c, then a < c.

q) is in the plane.

r) then there exists at least two different points on the line.

s) we have exactly one plane containing those three points.

t) For every real number a, there exists a unique real number (–a) such that a + (–a) = 0.

u) Mathematical expression which contains a placeholder symbol, but does not contain a relation symbol.

v) For three distinct coplanar rays, AB, AC, and AD with coordinates b, c, and d respectively, AC is between AB and AD if and only if b < c < d or d < c < b.

w) b < a

x) mAB + mBC = mAC

y) There is one and only one line through the point which is parallel to the given line.

z) One of the two sets of points in space on either side of a plane, called the separation plane.

y

w

z

c

x

k

t

o

a

g

p

r

q

j

( ( (

NameUnit II, Fundamental Terms, Unit Test Form C—Continued, Page 3—


For problems 27-34, use the figure shown.

27. Write three names for the line pictured. ___________________________

28. Name the ray that is opposite NP. ____________

29. Is it correct to say that point Q lies between points M and N? Why or Why not? _______________________________________________

30. Is point M on NP? ____________

31. Is point M on NP? ____________

32. Name three line segments that are in the line pictured. ______________________________

33. If MN = 7, NP = 3x + 5, and MP = 18, what is the value of “x”? Write an equation. ________________

34. What property allows you to set up the equation you used in exercise 33? ________________________

(B-3)

(B-4)

(B-4)

(B-4)

(B-4)

(B-5)

(C-7)

(C-7)

MN, NP, MP

NM

no; not collinear

no

yes

J Q

PM

N

HG

CB

AF D

P

TV

Q

R

S

D

CA

E

M

B

M PN

Q

0 1 2 3123

A B C D E F G

G IH F AO

BE

DC

Y

XV

Z

P

W

A

B

Q

X

C

Y

J

K

180 0

160 0

120 0

90 0

200

00

MN, NP, MP

x = 2

Segment Addition Property

7 + 3x + 5 = 18

3x +12 = 18

3x = 6

x = 2

( )



Use the number line below for problems 35-42.

35. Find DE 36. Find AC 37. Find BF

38. Find AE 39. Show that C is between B and D.

40. Show that D is the midpoint of CF. 41. Show that F is not between C and E.

42. Show that C is not the midpoint of AD.

(C-7)

(C-7)

(B-5)

(B-5)

(C-7) (C-7)

(C-7)

(C-7)

G

CB

AF D

P

TV

S

D

CA

E

M

B

0 1 2 3123

A B C D E F G

G F AO

BE

DC

A

B

Q

X

C

Y

180 0

160 0

120 0

90 0

200

00

5

4-

2

4

3

4

-1 --11

4-1+

11

4

7

4

= 2 --3

22+

3

2

31

2

=

5

4-

-11

4

5

4+

11

4

4

=BC CD BD+ =

+ =

B,C, and D Collinear

1

2

3

2

4

2

CF=3 2- -1 = 3 =3

Midpoint of CF is 1

2

( )

-1+1

23 = -1+

3

2=

1

2

The Coordinat

⋅

ee of D is 2

4or

1

2.

*C, D, F are Collinear.

Froom C to D: 1

2- -1 =

1

2+1 =

3

2=

3

2

From D to F:

( )

2-1

2=

3

2=

3

2

Therefore, D is the

CD DF

CD DF

=≅

mmidpoint since points

are collinear and seggments are congruent.

CF = 3 2- -1 = 2+1 = 3

FE = 3

4

( )

2-11

4=

3

4=

3

4

CE = 21

4

5

4- -1( )) =

5

4+1 = 2

1

4

Points C, E, and F are collinnear.

However, CF FE CE+ ≠

+ ≠

.

33

42

1

4

AC = 7

4-1- -2

3

4= -1+2

3

4=

-

44

4+

11

4=

7

4=

7

4

CD = 3

2

2

4- -1 =( ) 2

4+1 =

2

4+

4

4=

6

4=

6

4=

3

2

Points A, C, annd E are collinear.

but, AC CD≠ .



Use the figure at the right to find the measures asked for in problems 52-56.

52. m/BOC ____________________

53. m/EOF ____________________

54. m/DOE ____________________

55. m/COD ____________________

56. m/EOB ____________________

Using the diagram at the right, and your powers of observation, determine if each arc in problems 57-60 is a minor arc, major arc, or semicircle.

57. BDE ____________________

58. CAE ____________________

59. AE ____________________

60. BCD ____________________

(C-8)

(C-8)

(C-8)

(C-8)

(C-9)

(C-9)

(C-9)

(C-9)

(C-8)

PM CB V

A

E

M PN

Q

0 1 2 3123

A B C D E F G

G IH F AO

BE

DC

Y

XV

Z

P

W

A

B

Q

X

C

Y

J

K

180 0

160 0

120 0

90 0

200

00

J Q

PM

N

HG

E

CB

AF D

P

TV

Q

R

S

D

CA

E

M

B

M PN

Q

0 1 2 3123

A B C D E F G

G IH F AO

BE

DC

Y

XV

Z

P

W

A

B

Q

X

C

Y

J

K

180 0

160 0

120 0

90 0

200

00

((

((

90 20- or 70O

180 160- or 20O

160 120- or 40O

120 90- or 30O

160 20- or 140O

semicircle

major arc

minor arc

minor arc



In the diagram at the right, m/WPX = 38O, m/ZPY = 28O,and WZ and XV are diameters. Use this information to find the measure of the arcs given in problems 61-64.

61. mYZ ____________________

62. mVZ ____________________

63. mWZX ____________________

64. mWZY____________________

Answer problems 65-68, referring to the figure at the right, and using the following information.If m/BCY = 3x, m/BCQ = 3x + 30, and m/QCX = 2x + 30.

65. Find the value of x. _________

66. Find mAX ____________________

67. Find mBQ ____________________

68. Find mXQY ____________________

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

(C-9)

G IH F

E

D

Y

XV

Z

P

W

A

B

Q

X

C

J

K

180 0

160 0

((

(

J Q

PM

N

HG

E

CB

AF D

P

M PN

Q

0 1123

A B C D E

G IH F AO

BE

DC

Y

XV

Z

P

W

A

B

Q

X

C

Y

J

K

180 0

160 0

120 0

90 0

200

00

28O

38O

322O

208O

45O

75O

180O

15O

3x + 3x + 30 + 2x + 30 = 180

8x + 60 = 180

8x = 120

x = 15

( )(

((

(



Extending Your Mind—Determine if the statements in problems 69-72 are true or false.

69. Two congruent circles will have congruent radii. ____________

70. Two circles with the same center will be congruent circles. ____________

71. If two central angles are congruent, then their corresponding minor arcs are congruent. ____________

72. If two minor arcs are congruent, then theircorresponding central angles are congruent. ____________

true

false

false

true

Name


Unit II, Fundamental Terms, Unit Test Form D—Continued, Page 2—

_______13. linear pair

_______14. line

_______15. Multiplicative Inverse Property

_______16. Distributive Property ofMultiplication Over Addition

_______17. central angle

_______18. If two different lines intersect,then the intersection is...

_______19. Commutative Propertyof Addition

_______20. For every plane in space, the isat least one point in space...

_______21. congruent angles

_______22. semi circle

_______23. dihedral angle

_______24. minor arc

_______25. center

_______26. half-line

(B-8)

(A-2)

(C-1)

(C-1)

(C-6)

(C-1)

(C-4)

(B-8)

(B-9)

(B-6)

(C-9)

(B-9)

(B-4)

(B-9)

n) A set of points, not all of which lie on the same line.

o) The union of two half planes with the same edge.

p) a + b = b + a

q) of complete rotation

r) Mathematical expression which contains a relation symbol, but does not contain a placeholder symbol.

s) Two angles with the same measure.

t) A unique line.

u) There exists a unique real number 0, such that for every real number a, a • 0 = 0

v) One of two sets of points on either side of the separation point on a line.

w) that is not on the plane.

x) a (b + c) = a • b + a • c

y) If a < b, then a + c < b + c

z) Symbols that group numbers together, or, more generally, symbols which indicate the associations of mathematics.

b

g

c

x

m

j

p

w

s

d

o

k

e

v

1

4

NameUnit II, Fundamental Terms, Unit Test Form D—Continued, Page 3—


For problems 27-32, refer to the figure below.

27. m/HKI + m/IKJ = m/_____________

28. Name the property which allows you to answer exercise 27. ___________________________

29. If /HKI > /JKI, then ________________ is the bisector of / _________________.

30. m/LKI = __________________, so /LKI is called a(n) __________________ angle.

31. /IKJ and /JKL form a ____________ ____________.

32. Since m/HKI + m/HKL = ______________, /HKI and /HKL are said to be _________________.

(C-8)

(C-8)

(B-8)

(B-7)

(B-8)

(B-8)

HKJ

Angle Addition Property

HKJ

180O

linear

J Q

PM

N

HG

E

CB

AF D

P

TV

Q

R

S

I LK

H

J

W UV

QY

D

C

BA

Q

T

R

N S

M

c

N a

b

d

A B0

F

C

D

E

40o

D

A

E

M

B

Y

XV

Z

P

W

38

28

A

B

Q

X

C

Y

3x x0

0

straight

KI

pair

180O supplementary



33. Indicate for each of the four statements below whether you can or cannot conclude them from the figure at the right?

a) U, V, and W are collinear _______________

b) /UVY is a right angle _______________

c) V is the midpoint of WV _______________

d) Q is in the interior of /WVY _______________

In problems 34-37, determine whether each of the statements given is a good definition. If not, tell why.Assume the terms in each statement are previously defined.

34. The points lie on the same line. ____________________________________________________________

35. Points are coplanar if and only if they lie in the same plane. ________________________

36. Skew lines do not intersect. _______________________________________________________________

37. A rhombus is a quadrilateral. _____________________________________________________________

38. How many planes are determined by three non-collinear points? ________________________

39. Suppose two different planes contain AB. Describe the intersection of the planes. ________________________________________________________________________________________________________

40. Two angles form a linear pair. The degree measure of one of the angles is three times the degree measure of the other angle. Find the degree measure of each angle. ___________________

(B-1)

(B-1)

(B-1)

(B-1)

(C-2)

(C-2)

(B-8)

PM

HG

CB

AF D

TV

S

I LK

H

J

W UV

QY

D

C

A

Q

T

R

N S

M

c

N a

b

d

A B0

F

C

D

E

40o

A

E

Y

XV

Z

P

W

38

28

A

B

Q

X

C

Y

3x x0

0

(B-5)

can

can

cannot

can

no; the set to which these points belong is not identified.

yes

no; the distinguishing qualities are not identified

no; the distinguishing qualities are not identified.

one

The intersection of the

two planes is the line containing line segment AB.

45O; 135O

3x + x = 180

4x = 180

x = 45

3x = 135

E

C

DP

TV

Q

R

S

UV

Y

D

C

BA

B0

F

C

D

40o

D

CA

E

M

B

3x x



In the three-dimensional figure at the right, AD DB, AD DC, and DC DB.

41. Name the dihedral angle with edge of, CD. ________________________

42. Name the edge of /B – AD – C. ________________________

43. DB is called the ________________________ of half-plane ADB and half-plane ______________.

For problems 44-50, use the diagram below, and find an example for each set of conditions given. Assume line c and line d are parallel.

44. Two non-coplanar lines. ______________________________

45. Two parallel lines. ___________________

46. Two segments that are not parallel. ___________________

47. Two coplanar lines that intersect. ___________________________

48. Two skew lines ___________________

49. A line which lies in plane M ___________________

50. Line a intersects plane M in point ___________________.

(B-3)

(B-3)

(B-3)

(B-3)

(B-3)

(B-2)

(B-3)

PM

G

CB

TV

I LK

H

J

W UV

QY

D

C

BA

Q

T

R

N S

M

c

N a

b

d

A B0

F

C

D

E

40o

D

A

E

M

Y

XV

Z

P

W

38

28

A

B

Q

X

C

Y

3x x0

0

(B-6)

(B-6)

(B-6)

A – CD – B

AD

edge CDB

line d and line a (or line a and line b)

line d and RQ

RT and SR

line d and line a (or line a and line b)

line d and line a

NS

R

J Q

PM

N

HG

E

CB

AF D

P

V

I LK

H

J

W UV

QY

A

Q

T

R

N S

M

c

N a

b

d

A B0

F

C

D

E

40o

Y

XV

Z

P

W

38

28

A

B

Q

X

C

Y

0

0



Using the diagram at the right, state whether you can reach each conclusion given in problems 51-60.

51. m/FOB = 50 _______________

52. m/AOC = 90 _______________

53. m/DOC = 180 _______________

54. AO = OB _______________

55. /AOC > /BOC _______________

56. m/AOF = 130 _______________

57. Points A, O, and B are collinear. _______________

58. Point C is in the interior of /AOF _______________

59. /AOE and /AOD are adjacent angles. _______________

60. OA and OB are opposite rays. _______________

I LK

J

W UV

Q

A

Q

T

R

N S

M

c

N a

b

d

A B0

F

C

D

E

40o

Y

XV

Z

P

W

38

28

A

B

Q

X

C

Y

3x0

0

(C-8)

(C-8)

(C-8)

(B-5)

(B-8)

(B-8)

(B-2)

(B-8)

(B-9)

(B-4)

yes

yes

yes

no

yes

yes

yes

yes

no

yes

Documents

Geometry: A Complete Course - VideoText · examples are considered,in noticeable detail, taking students, once again, through the logic of the concept development process.The premise