Chapter 1: Points, Lines, and Planes - Mr. Jaime … 1: Points, Lines, and Planes Point, line, plane – are undefined terms. They do not need to be defined

2014-15 Geometry Semester 1 Final Exam Study Guide FCS, Mr. Garcia 12/8/14

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Name_________________________________ Date___________________ Period______ This is your semester 1 exam review study guide. It is designed for you to do a portion each day until the day of the exam on Thursday of next week. Topics that we have covered on chapters 1 through 4 are outlined below for your review. =============================================================================== Chapter 1: Points, Lines, and Planes Point, line, plane – are undefined terms. They do not need to be defined. Definitions or defined terms are explained using undefined terms and/or other defined terms. Space is defined as a boundless, 3-dimensional set of all points. Space can contain lines and planes.

1. How do you name a line? __________, a line segment? __________, a ray? ___________, a plane? __________,

an angle, ____________, a triangle ? ________, a quadrilateral? _________, a pentagon? _________.

2. What does it mean for 3 or more points to be collinear? _______________________________, Noncollinear? ____________________________________.

3. What does it mean for 3 or more points to be coplanar? _______________________________, Noncoplanar? _______________________________.

Relationships of lines and planes:

4. What does it mean for 2 lines to be parallel? ______________________________________. 5. What is the symbol for parallel? ______ What is the symbol for perpendicular? ______. 6. When 2 lines intersect, they intersect at a p ________________. 7. When a line and a plane intersect, they intersect at a p ____________________. 8. When 2 planes intersect, they intersect at a l __________________________. 9. An angle bisector could be a s_________________, a l____________________, or a r____________. 10. Any segment, line, or plane that intersects a segment at its m_________________ is called a

Segment b______________________. 11. When a line segment, a ray, or a line, bisects a segment, the bisector creates two

s__________________ that are equal in m____________________, or equal in l________________.

12. If 2 segments are equal in length, or in measure, then they are said to be c____________________. The postulate that states this is called? _____________________. Look it up in you textbook.

13. When a line segment, a ray, or a line, bisects an angle, it creates two c__________________ angles,

and their measures are e_________________.

14. What is the difference between an expression and an equation? Write an example of each.


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15. A point where the 2 sides of an angle meet is called? V____________.

16. The slope of a line yx

ΔΔ

can be calculated when you are given 2 p________________.

17. What is the slope formula? (Look it up in your textbook if you don’t remember)

18. The Pythagorean theorem formula and the distance formula are really the same, however you use the Pythagorean formula, c2 = a2 + b2 , when you are given 2 d_____________________, and you use the

distance formula, (x2 − x1)2 + ( y2 − y1)2 , when you are given 2 p____________. Write 2 example problems to show their use.

19. The midpoint of a segment is the point halfway between the e______________ of a segment.

20. You use the midpoint formula (pg. 27), ' '( , )2 2

sum of x s sum of y s or 1 2 1 2( , )2 2

x x y y+ + , when you are

given 2 p________________, and you are asked to find the midpoint of a s_______________________.

Create a problem example and solve it. Find the slope of the line through the given points.

21. A(-3,8), B(4,2) _____

22. What is the slope of any line parallel to the line through points A and B? _____

23. What is the slope of any line perpendicular to the line through points A and B? _____ 24. C(1,-3), D(9,-9) _____

25. E(-2,-3), F(-6,-5) _____


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============================================================================== Chapter 1-1 exercises. Refer to the figure to the right to answer problems 1 - 7. _______ 1. The line intersecting plane P.

______2. The intersection of AC and XF .

______3. Are points B, F, and X collinear?

______4. Are points A, B, and X coplanar?

______5. Are points A, B, and X contained in Plane P?

____ ____ ____6. Identify 3 non-collinear points

____ ____ ____ ____7. Identify 4 non-coplanar points. Use the midpoint theorem, the segment addition property, or the distance formula to solve the following problems. 8. If B is the midpoint of AC and AB = 2x – 3 and BC = 5x – 24, find x, AB, and BC. X = _____, AB = _____, BC = _____ 9. If XB = 14 and XF = 20, find BF. ______ 10. If B is the midpoint of XF and XB = x + 11 and BF = 5x – 1, find x and XF. ____, ____ 11. If AB = 3x, BC = x + 2, and AC = 38, find x and AB. _____, _____ 12. If the coordinate x of G is –8 and the x coordinate of H is 9, find GH. ______ 13. Find the midpoint of the segment having the given endpoints: a. A(-2, -4), B(3, 8) ______ b. C( 3, -4), D( -3, -1) ______ c. E( 2, 1), F(5, 1)_____ 14. Find the distance between the given endpoints: a. A(-2, -4), B(3, 8) ______ b. C( 3, -4), D( -3, -1) ______ c. E( 2, 1), F(5, 1)_____ d. If the length of PQ is twice the length of AB , then find PQ. _____ e. If the length of RS is one third the length of EF , then find RS. _____ 15. Find the coordinates of A, the missing endpoint, if B(-2, 5) is the midpoint of AC , and the coordinates of C are (-5, 4). See example 5 on page 28.

j

X

C B A

P D F


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Also do, Pg. 79-80: 2-22 (even); (more practice exercises). ============================================================================== Chapter 1-4, Pg. 36 Angle Measure

1. An angle is formed by two noncollinear rays that have a common endpoint. The rays are called s___________ of the angle. The common endpoint is the v_________________ of the angle and it must always be in the center of the name of the angle. Angles are measure in d_____________.

2. There 3 types of angles: a r____________ angle; it measures ________________ degrees. 3. An a_______________ angle; it measures < 90 degrees, and an o____________ angle; it measures

_________ degrees.

4. One could say that there is a fourth type of angle called the straight angle, which is just a line made up of two opposite rays; it measures 180 d__________________.

1-5 Angle pair relationships

1. Adjacent angles are 2 angles that have a common v__________ and a common s___________, but no common interior points. Draw an example of 2 adjacent angles and a counterexample.

2. A linear pair is a pair of adjacent angles with noncommon sides that are opposite r_________. Draw an example of a linear pair and a counterexample.

3. Vertical angles are two nonadjacent angles formed by two intersecting lines. Draw an example of vertical angles and a counterexample.

4. Complementary angles are two angles, whose m___________________ add up to 180 d________________. Draw an example.

5. Supplementary angles are two angles, whose m___________________ add up to 180 d________________. Draw an example.

6. Perpendicular lines intersect to form f__________ right a________________. Draw a picture that illustrates this. Add the right angle symbol to your drawing. The symbol of perpendicular is ______.


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============================================================================= Refer to Figure 2. Matching, you may use more than one letter to describe the angle(s). ________ 1. ∠1 and ∠2 ________ 2. ∠1 and ∠5 ________ 3. ∠3 and ∠4 ________ 4. ∠1 and ∠BOE ________ 5. ∠1 and ∠6 ________ 6. ∠AOF and ∠BOE ________ 7. ∠AOC and ∠COE ________ 8. ∠2 and ∠5 ________ 9. ∠4 and ∠AOD ============================================================================= Refer to figure 2 to solve problems 10 - 17. 10. If m∠3 = 27°, then m∠4 = _____, and m∠1 + m∠BOD = m∠_____. 12. If m∠1 = 46° and m∠4 = 59°, then m∠DOF = _____. 13. If OD bisects ∠COE, then m∠4 = _____. 14. If OD ⊥ BF , then m∠4 + m∠5 = _____. 15. If OD ⊥ BF and m∠4 = 65°, then m∠1 = _____, and m∠2 = _____, m∠6 = _____, m∠AOF = _____. 16. If OD ⊥ BF , name all the pairs of complementary angles.______________ _____________________________________________________________ 17. If OD is the ⊥ bisector of BF , which segments are congruent? __________ 18. Name the vertex of ∠DOF _____________. 19. Write another name for ∠6 ________________. ============================================================================= Refer to figure 3 to solve problems 18 - 21. 20. Given: m∠2 = 9x +28 and m∠3 = 47 – 2x, x = _____, m∠2 = _____ 21. Given: m∠1 = 3x + 5 and m∠3 = 65, x = _____ 22. Given: m∠2 = 9x +2 and m∠4 = 7x + 36, x = _____, m∠2 = _____ 23. Given: m∠1 = x-9 and m∠2 = 2x, x = _____, m∠1 = _____

Figure 2

A

B C D

E

F G

1

2 3 4

5 6 O •

•

• •

•

• •

•

a. acute angles b. right angles c. obtuse angles d. adjacent angles e. linear pair f. complementary angles g. supplementary angles h. vertical angles i. congruent angles

Figure 2

A

B C

D

E

F G

1

2 3 4

5 6 O •

•

• •

•

• •

•

1 2

3 4

Figure 3


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Do problems page 80-81: 24 – 30 (even). 1-6 and 1-7 Two and Three Dimensional Figures

1. A polygon is a closed figure formed by a finite number of c_____________ segments called s___________ such that the sides have a common e_______________ are noncoplanar, and each side intersects exactly 2 other sides, but only at their e_________________.

2. The vertex of each angle is a vertex of the polygon. A polygon is named by the letters of its

v____________. Written in the order of the consecutive v________________.

3. A polygon can be c_________________ and convex.

4. A polygon with 4 sides is called a q__________________. One with five sides is called a p_________________. One with n-sides is called an n-gon. In the name Polygon, poly stands for m__________ and gon stands for s________________.

5. An equilateral polygon is a polygon in which all s_____________ are congruent, and an equiangular polygon is one in which all a______________________ are c______________________.

6. A convex polygon that is both equiangular and e__________________ is called a r______________ polygon.

7. The perimeter of a polygon is the s________ of the lengths of the s____________. The circumference of a circle is the d_________________ around the circle.

8. The area of a figure is the number of square units needed to cover a s_______________. Review all the formulas on page 58 in your textbook. Draw the figure and write the formula underneath it.

9. Dasan has 32 feet of fencing to fence in a play area for his dog. Which shape of play area uses the most or all of the fencing and encloses the largest area?

a. Circle with radius of about 5 feet b. Rectangle with length 5 feet and width 10 feet c. Right triangle with legs of length 10 feet each d. Square with side length 8 feet

10. Find the perimeter and area of ABC with vertices A(-1, 4), B(-1, -1), and C(6, -1).

11. A rectangle of area 360 sq. meters is 10 times as long as it is wide. Find its length and width.


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12. The vertices of a rectangle with side lengths of 10 and 24 units are on a circle of radius 13 units. Find

the area between the figures. See page 67 to review 3-Dimensional figures.

1. A solid figure with all flat surfaces that enclose a single region of space is called a p________________. Each flat surface or face is a polygon. The line segments where the faces intersect are called e_________. The point where the 3 or more edges intersect is called a v________________.

2. A prism is a polyhedron with two parallel congruent f____________ called b____________ connected

by parallelogram faces. Draw one example.

3. A pyramid is a polyhedron that has a polygonal b______________ and 3 or more triangular f_________ that meet at a common v_______________ (peak). Draw one example.

4. A cylinder is a solid with congruent parallel circular b____________ connected by a curved

s_____________. Draw a picture.

5. A cone is a solid with a circular base connected by a curved s________________ to a single v____________. Draw a picture

6. A sphere is a set of points in space that are the same distance from a given p____________. A sphere

has no faces, no e________, and no v______________. Draw a picture.

7. Find the volume of a cube that has a total surface area of 54 square millimeters. See page 69 for formulas of following 3-D figures:

8. Prism, regular pyramid, cylinder, cone, and sphere. Draw a picture of each figure listed and write the formulas for volume and surface area underneath them.


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9. Do problems on page 81-82: 32-43 (all).

A word problem having to do with the equation of a line. It’s the end of the semester, and the clubs at school are recording their profits. The Science Club started out at $20 and has increased its balance by an average of $10 per week. The Math Club saved $5 a week and started out with $50 at the beginning of the semester. a) Define x and y to fit the problem. b) make a table of values for each club. c) Write an equation for each club. d) Draw a complete graph for each rule and the same axes. e) When do the clubs have the same balance? Show how you can get this number both with the graph and with the equations in c above. f) What is the balance at that point?


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============================================================================================= Chapter 2 Logic & Reasoning Terms:

1. Deductive r____________________ uses facts, rules, definitions, theorems, or properties to reach logical conclusions.

2. A counterexample is a f________________ example about a conjecture, it could be a number, a drawing or a statement.

3. I___________________ reasoning uses a pattern of examples or observations to make a conjecture, or an intelligent guess.

4. Conjecture is a conclusion reached by using i______________________ reasoning. The h__________________ is the if part of a conditional statement, and the then part is its c__________________________.

5. A conditional statement is a statement that can be written in if-then form. The if part of a conditional statement is called the h_____________________, and the then part is called the conclusion.

6. The converse of a conditional is formed by exchanging the hypothesis and c____________________. 7. The inverse is formed by negating both the h___________________ and the

c_____________________. 8. The contrapositive is formed by n_________________________ both the hypothesis and the

conclusion of the converse statement. 9. A postulate or axiom is a statement that is accepted as true without p________________. 10. A theorem is has to be p_____________________ for it to be accepted as true. 11. Go to page 127 to read and memorize postulates 2.1 to 2.7. Go to page 144 to read/memorize postulates

2.8 and 2.9. 12. A 2-column proof is a proof base on deductive r______________________. 13. See page 145 or Theorem 2.2 Properties of Segment Congruence. 14. Page 151 Postulate 2.10 Protractor Postulate 15. 2.11 Angle Addition Postulate states: m∠ABD +m∠DBC = m∠ABC . Draw a figure that matches

the postulate description. Read the following instruction carefully to fill in the blanks below. Each Instruction after the letters corresponds with the letters under each number. A. Restate each of the following given statement into an “if-then” statement. B. Underline the hypothesis and circle the conclusion. C. Is the statement true or false? Circle your answer. D. Write the converse of the conditional and determine whether it is true or false. E. Write the inverse of the conditional and determine whether it is true or false. F. Write the contrapositive of the conditional and determine whether it is true or false. G. If possible, write the bi-conditional statement in “if and only if” form. If not, write a counter example

demonstrating why not. 1. Tardy students receive detention. A. & B. ____________________________________________________________ C. T or F D. ___________________________________________________________________T or F E. ___________________________________________________________________T or F


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F. ___________________________________________________________________T or F G. ______________________________________________________________________ 2. All right angles are congruent. A. & B. ____________________________________________________________ C. T or F D. ___________________________________________________________________T or F E. ___________________________________________________________________T or F F. ___________________________________________________________________T or F G. ____________________________________________________________________ 3. A triangle is a polygon that has three sides. A. & B. ____________________________________________________________ C. T or F D. ___________________________________________________________________T or F E. ___________________________________________________________________T or F F. ___________________________________________________________________T or F G. ______________________________________________________________________ 4. Supplementary angles are two angles whose sum is 180°.

A. & B. ____________________________________________________________ C. T or F D. ___________________________________________________________________T or F E. ___________________________________________________________________T or F F. ___________________________________________________________________T or F G. _____________________________________________________________________


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Chapter 3 Topics: parallel Lines & Their Relationships =============================================================== Terms: If 2 parallel lines are cut by a transversal, then corresponding angles are ≅ , alternate i_____________________ angles are ≅ , alternate e_____________________ angles are ≅ , consecutive i___________________ angles are s__________________________ parallel lines never intersect, parallel lines have the same s_________________________ =========================================================================== Refer to figure 4 to determine which lines if any are parallel. 1. Given: ∠1 ≅ ∠5 _____ 2. Given: ∠8 ≅ ∠12 _____ 3. Given: ∠7 ≅ ∠13 _____ 4. Given: ∠4 ≅ ∠14 _____ 5. Given: ∠6 ≅ ∠11 _____ 6. Given: ∠10 ≅ ∠15 _____ 7. Given: ∠3 and ∠13 are supplementary _____ Given a b, l m . (Refer to figure 4) 8. If m∠ 12 = 67 , then m∠ 3 = ______ 9. If m∠ 6 = 108 , then m∠ 16 = ______ 10. If m∠ 4 = 123 , then m∠ 10 = ______ 11. If m∠ 1 = 71 , then m∠ 10 = ______ 12. m∠1 = 2x + 7 and m∠16 = x + 30, x = _____, m∠1 = _____, m∠16 = _____ 13. m∠11 = 3x + 6 and m∠13 = x + 26, x = _____, m∠11 = _____, m∠13 = _____ 14. m∠2 = 11x - 16 and m∠7 = 7x + 28, x = _____, m∠2 = _____, m∠7 = _____

===============================================================

Chapter 4 Topics: Triangle Relationships Term: Triangles may be classified by angles: r___________ triangle, a______________ triangle, o_____________________ triangle. Equiangular triangles contain all 60° angles, and an eq_______________________ contains all sides congruent. An i_________________ triangle has at least 2 sides e______________________. A scalene has no s_____________ equal. All angles of a triangle add up to ____________ degrees. The external theorem states that the e___________________ angle is equal to the 2 r____________________ angles of a triangle. ============================================================== Find the value of x. 1. x = _______ 2. x = _______ 3. x = _______

1 2 3

4

5 6

7 8 9 10

11 12

13

14 15

16

a b

l

m

Figure 4

100°

x°

70°

x°

70°


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In ΔABC, find x and m∠A, then classify the type of triangle according to sides and angels. 4. 246 −=∠ xAm , 72 −=∠ xBm , and 4+=∠ xCm x = ______, =∠Am Triangle type__________________________ 5. m∠A = 8x + 9, m∠B = 3x – 4, m∠C = 9x + 15 x = ______, =∠Am Triangle type ____________________ Using the given information, classify each triangle according to its sides and angles. 6. DFZΔ , DF < DZ and m∠ D = 90. 10. MNOΔ , 27=∠Mm and 82=∠Om .

7. AWVΔ , AW = AV and m∠A < 90. 11. LJRΔ , 35=∠Lm and 104=∠Rm

.

8. PONΔ , PO = 5, ON = 5, PN = 5. 12. KMNΔ , Mm∠ >90°, MN = MK.

9. LJIΔ , 45=∠Lm and 90=∠Im . 13. SYXΔ , Sm∠ = 60° and Ym∠ = 60°. Use the distance formula to classify the triangle by the measure of its sides. 14. A(1, 0) B(3, 3) C(2, 4) AB = _____ BC = _____ AC = _____ Classification ________________ 15. D(4, -6) A(-2, 5) V(0, 7) DA =_____ AV = _____ DV = Classification: _______________

x


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=============================================================== Chapter 4 Topic: Congruent Triangles Term: constructions of congruent triangles, 2 sides and the included angle are ≅ (SAS), 2 angles and the included side are ≅ (ASA), three congruent sides are ≅ (SSS), 2 angles and the non-included side are ≅ (AAS), the hypotenuse and a leg of a right triangle (HL) =============================================================== Identify the congruent triangles and justify your answer. If congruency can not be proven write “n p” in both blanks. 1. Given: FDCAandEFBCEDAB ≅≅≅ ,, ΔBAC ≅ Δ__________ by _____________. 2. Given: MPandMPMPMTSM ,, ≅≅ bisects ∠SMT

ΔMPS ≅Δ__________ by _____________.

3. Given: ,MNOM ⊥ ,PQPR⊥ ,PRMO ≅ and RQON ≅

ΔMNO ≅ Δ_________ by ______________.

4. Given: HKGHJKFG ≅≅ , ΔHJK ≅ Δ_________ by _______________.

5. 6.

A B

C

D

F

E

M

S P T

M

P

O

R

Q

N

F

J

H

G

K

Given: C is the Midpoint of ΔABC ≅ Δ _______by _________________

A

C

E

B

D

Y Z

W

X

Given: bisects ∠YXW, ∠YZX is a right angle. ΔXYZ ≅ Δ ________ by ________________


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For the following problems, ΔABC ≅ ΔDEF. 7. Given: AB = 3y + 12, DE = 5y – 18, find DE. ______ 8. Given: m∠C = 4y – 23, m ∠F = 2y – 5, find the m∠C. ______ 9. Use the distance formula to determine whether the triangles with the given vertices are congruent. Given: ∆PQR : P(1,2), Q(3,6), R(6,5) ∆ KLM : K(-2,1), L(-6,3), M(-5,6) PQ = KL = QR = LM = PR = KM = Are they Congruent? Why? ============================================================== 10. Proofs:

1. Given: a b, n m

2. Prove: 4 10∠ ≅ ∠

Statements Reasons

1.

2.

3.

1

a

6 5 1

10

16

4

2

11

8 7

15

92

n m

b

13 14

2

a


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11. Given : AD BC ; AD ≅ BC

Prove: ∆ ABD ≅ ∆ CDB Statements Reasons

1. 2. 3. 4. 5. 13. Use the graph to the right and use the

a) Pythagorean Theorem to determine the length of the longest segment.

Round to the nearest hundredth. Be sure to indicate the segment.

b) List the segments order from least to greatest. 14. Use the graph to the right to answer the following questions. State the coordinates for an endpoint of the segment with point B as one endpoint and point A as a midpoint. 15. Given: C is the midpoint of BD , BC = (2x – 3)cm and CD = (5x – 24)cm. Find the length of BD . 16. Find the value of x in the figure.

A B

C D 1

2

3 4

•B

•C

•F

•D • A

3x + 4

2x + 1


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17. If m∠FBC = 74 and m∠ 1 = 3x - 8 and m∠ 2 = 5x + 26, find x and m∠ 3.(4 points) 18. If m∠ 1 = °41 , and m∠ DOF = °87 , what is m∠ 4? 19. If m∠ 3 = 8x – 12 and m∠ 4 = 4x + 6, and m∠ 1 = 3x – 9, find m∠ 1. 20. If ∠ 3≅ ∠ 4, then OD

is a(n) ______________.

21. If GA

// BF

, their slopes are _______. 22. If point B and point D are equidistant from AE

, what conclusion

can be made about ∠ 1 and ∠ 4? 23. What is the sum of ∠ 1, ∠2, ∠3, ∠4, ∠5, and ∠6? =============================================================== 24. KNGΔ is an isosceles triangle with K∠ as the vertex angle, and 5 2KN x= − , and 2 4GK x= + .

a. Draw a diagram and label the angles and the sides with their lengths in algebraic form.

b. What is the length of KN ? …Of KG ? c. For what range of values for GN will the lengths still form a triangle ? d. Make a table of lengths possible for NG . (Use only integers) e. Using the range of values above, find 1 value that will form an ACUTE triangle.

Justify using the Pythagorean theorem. f. Using the range of values above, find 1 value that will form an OBTUSE triangle.

Justify using the Pythagorean theorem. ============================================================== 25. In ΔQRT, the angles listed from largest to smallest are:

a) ∠ Q , ∠R , ∠T b) ∠ R , ∠Q , ∠T

c) ∠ T , ∠R , ∠Q d) ∠ Q , ∠T , ∠R

A

F

C

D

B 1

2 3

A

B

E

F G

1

Figure 2

C D

2 3 4

5 6 O •

•

• •

•

• •

•

T R

Q

25 19

30


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Why are the following triangles are congruent? Justify your reasoning! Be sure to use the phrase “two sides and the included angle are congruent” instead of SAS! 26. Given : AB ≅ CD; AD ≅ BC 27. Given: AE ≅ BC ; ∠E ≅ ∠C Prove : ∆ABD ≅ ∆ CDB D is the midpoint of EC Prove: ∆ADE ≅ ∆BDC 28. Determine which postulate can be used to prove the triangles are congruent. If the triangles cannot be proven congruent write NONE. Be sure to write out the postulate (EX: 2 sides and the included angle are congruent instead of SAS)

A B

C D A B

E D C

Documents

Chapter 1: Points, Lines, and Planes - Mr. Jaime … 1: Points, Lines, and Planes Point, line, plane – are undefined terms. They do not need to be defined