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Holt Geometry Review for Mastery solution guide.
Holt CaliforniaGeometryReview for Mastery WorkbookTeachers GuideCopyright by Holt, Rinehart and Winston. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Teachers using HOLT CALIFORNIA GEOMETRY may photocopy complete pages in sufficient quantities for classroom use only and not for resale.HOLT and the Owl Design are trademarks licensed to Holt, Rinehart and Winston, registered in the United States of America and/or other jurisdictions.Printed in the United States of AmericaIf you have received these materials as examination copies free of charge, Holt, Rinehart and Winston retains title to the materials and they may not be resold. Resale of examination copies is strictly prohibited.Possession of this publication in print format does not entitle users to convert this publication, or any portion of it, into electronic format.ISBN 13: 978-0-03-099026-7ISBN 10: 0-30-099026-21 2 3 4 5 6 7 8 986210 09 08 07Copyright by Holt, Rinehart and Winston.i i i Holt GeometryAll rights reserved.ContentsChapter 1 .......................................................................................................................................1Chapter 2 .......................................................................................................................................4Chapter 3 .......................................................................................................................................8Chapter 4 .....................................................................................................................................11Chapter 5 .....................................................................................................................................15Chapter 6 .....................................................................................................................................19Chapter 7 .....................................................................................................................................22Chapter 8 .....................................................................................................................................25Chapter 9 .....................................................................................................................................28Chapter 10 ...................................................................................................................................31Chapter 11 ...................................................................................................................................35Chapter 12 ...................................................................................................................................38Copyright by Holt, Rinehart and Winston.1Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.1Holt GeometryAll rights reserved.1-1LESSONReview for MasteryUnderstanding Points, Lines, and PlanesA point has no size. It is named using a capital letter. All the figures below contain points.Figure Characteristics Diagram Words and Symbolsline 0 endpoints extends forever in two directions! "line AB or __ AB line segment or segment2 endpoints has a finite length8 9segment XY or_XY ray 1 endpoint extends forever in one direction1 2ray RQ or ___ RQ A ray is named starting with its endpoint.plane extends forever in all directions&'( plane FGH or plane VDraw and label a diagram for each figure.1.point W2.line MN W
- .3._JK4. __ EF * + % &Name each figure using words and symbols.5. CD6. 43 line CD or ___ CD ray ST or ___ ST 7.Name the plane in two different ways.8. 78 ,-.
plane LMN; plane Q segment WX ;_WX Ppoint PCopyright by Holt, Rinehart and Winston.2Holt GeometryAll rights reserved.1-1LESSONTerm Meaning Modelcollinear points that lie on the same line&'(F and G are collinear.F, G, and H are noncollinear.noncollinear points that do not lie on the same linecoplanar points or lines that lie in the same plane:789W, X, and Y are coplanar.W, X, Y, and Z are noncoplanar.noncoplanar points or lines that do not lie in the same planeFigures that intersect share a common set of points. In the first model above, __ FH intersects __ FG at point F. In the second model, __ XZ intersects plane WXY at point X.Use the figure for Exercises 914. Name each of the following.*#0!$+"
9.three collinear points10.three noncollinear points Possible answers: A, P, and B; Sample answer: A, P, and D 11.four coplanar points12.four noncoplanar points Sample answer: C, P, B, and D Sample answer: J, D, P, and B 13.two lines that intersect ___ CD14.the intersection of __ JK and plane R ___ AB and __ JK point DReview for MasteryUnderstanding Points, Lines, and Planes continuedC, P, and D; J, D, and KCopyright by Holt, Rinehart and Winston.3Holt GeometryAll rights reserved.LESSON LESSON1-2Review for MasteryMeasuring and Constructing SegmentsThe distance between any two points is the length of the segment that connects them. centimeters (cm)0 1 2 3 4 5 6 7% & ' ( *The distance between E and J is EJ, the length of_EJ . To find the distance, subtract the numbers corresponding to the points and then take the absolute value.EJ 7 1 6 6 cm Use the figure above to find each length.1.EG2.EF3.FH 4 cm 1.5 cm 3 cm001 12 022 1 X On_PR , Q is between P and R. If PR 16, we can find QR. PQ + QR PR 9 x 16x 7QR 74. * + ,
Y5. ! " #
ZFind JK.2Find BC.66. 3 4 6
NN7. 7 8 9
A AFind SV.41Find XY.218. $ % & X9. 3 4 5
YYFind DF.135Find ST.22Copyright by Holt, Rinehart and Winston.4Holt GeometryAll rights reserved.LESSON1-2Review for MasteryMeasuring and Constructing Segments continuedSegments are congruent if their lengths are equal.AB BCThe length of_AB equals the length of_BC . _AB _BC _AB is congruent to_BC .Copying a SegmentMethod Stepssketch using estimation Estimate the length of the segment. Sketch a segment that is about the same length.draw with a ruler Use a ruler to measure the length of the segment. Use the ruler to draw a segment having the same length.construct with a compass and straightedgeDraw a line and mark a point on it. Open the compass to the length of the original segment. Mark off a segment on your line at the same length.Refer to triangle ABC above for Exercises 10 and 11. 10.Sketch_LM that is congruent to_AC .11.Use a ruler to draw_XY that is congruentto_BC . 12.Use a compass to construct_ST that is congruent to_JK . *+ 3 4The midpoint of a segment separates the segment into two congruent segments. In the figure, P is the midpoint of_NQ .
.X X0 1 13. _PQ is congruent to _NP or_PN . 14.What is the value of x? 4 15.Find NP, PQ, and NQ. 12, 12, 24 ! #"
, -8 9Copyright by Holt, Rinehart and Winston.2Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.5Holt GeometryAll rights reserved.LESSONReview for MasteryMeasuring and Constructing Angles 1-3An angle is a figure made up of two rays, or sides, that have a common endpoint, called the vertex of the angle.89
:There are four ways to name this angle.YUse the vertex.XYZ or ZYXUse the vertex and a point on each side.2Use the number.Name each angle in three ways.1.0 21
2.(+*Q, PQR, 1 J, HJK, KJH3.Name three different angles in the figure. $!#"ABD, ABC, DBCAngle acute right obtuse straightModelAAAAPossible Measures0 a 90 a 90 90 a 180 a 180Classify each angle as acute, right, obtuse, or straight.4.NMP .1-0, obtuse5.QMN right6.PMQ acuteThe vertex is Y.The sides are __ YX and __ YZ .Copyright by Holt, Rinehart and Winston.6Holt GeometryAll rights reserved.LESSON1-3Review for MasteryMeasuring and Constructing Angles continuedYou can use a protractor to find the measure of an angle. '%$ &1OO8O11O7O12OGO18O5O 14O 4O 15O 8O1GO 2O17O1O8O1OO7O11O GO12O5O18O4O14O8O15O2O1GO1O17OOOUse the figure above to find the measure of each angle.7. DEG8. GEF55 125The measure of XVU can be found by adding. 6875
mXVU mXVW mWVU 48 48 96Angles are congruent if their measures are equal. In the figure, XVW WVU because the angles have equal measures. ___ VW is an angle bisector of XVU because it divides XVU into two congruent angles.Find each angle measure.$%
!"#&9.mCFB if AFC is a straight angle.10.mEFA if the angle is congruent to DFE.102 51 11.mEFC if DFC AFB.12.mCFG if __ FG is an angle bisector of CFB.129 51DEG is acute.GEF is obtuse.Copyright by Holt, Rinehart and Winston.7Holt GeometryAll rights reserved.LESSONReview for MasteryPairs of Angles 1-4Angle PairsAdjacent Angles Linear Pairs Vertical Angleshave the same vertex and share a common sideadjacent angles whose noncommon sides are opposite raysnonadjacent angles formed by two intersecting lines
1 and 2 are adjacent. 3 and 4 are adjacent and form a linear pair. 5 and 6 are vertical angles.Tell whether 7 and 8 in each figure are only adjacent, are adjacent and form a linear pair, or are not adjacent.1.
2.
3. adjacent and form a linear pair only adjacent not adjacentTell whether the indicated angles are only adjacent, are adjacent and form a linear pair, or are not adjacent.4.5 and 4 only adjacent
5.1 and 4 not adjacent6.2 and 3 adjacent and form a linear pairName each of the following.7.a pair of vertical angles Possible answers:1 and 6, 2 and 5
8.a linear pair Possible answers: 1 and 2; 1 and 5; 5 and 6; 6 and 29.an angle adjacent to 4 3Copyright by Holt, Rinehart and Winston.8Holt GeometryAll rights reserved.LESSON1-4Review for MasteryPairs of Angles continuedAngle PairsComplementary Angles Supplementary Anglessum of angle measures is 90 sum of angle measures is 180
m1 m2 90In each pair, 1 and 2 are complementary. m3 m4 180In each pair, 3 and 4 are supplementary.Tell whether each pair of labeled angles is complementary, supplementary, or neither. 10. 11.
complementary neitherFind the measure of each of the following angles. 12.complement of S 34 3
13.supplement of S 124 14.complement of R 68 2
15.supplement of R 158 16.LMN and UVW are complementary. Find the measure of each angle if mLMN (3x 5) and mUVW 2x. mLMN 56; mUVW 34Copyright by Holt, Rinehart and Winston.3Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.9Holt GeometryAll rights reserved.LESSONReview for MasteryUsing Formulas in Geometry 1-5The perimeter of a figure is the sum of the lengths of the sides. The area is the number of square units enclosed by the figure.Figure Rectangle SquareModelW W
SSSSPerimeter P 2 2w or 2( w) P 4sArea A w A s 2Find the perimeter and area of each figure.1.rectangle with 4 ft, w 1 ft2.square with s 8 mm 10 ft; 4 ft2 32 mm; 64 mm23. CM4.ININX X 28 cm; 49 cm2 (24 2x) in.; 12x in2The perimeter of a triangle is the sum of its side lengths. The base and height are used to find the area.HACB BCHA PerimeterAreaP = a + b + cA =1 __ 2 bh orbh ___ 2
Find the perimeter and area of each triangle.5. FTFTYFT6. 9 cm6.7 cm6 cm8.5 cm (18 y) ft; 4y ft2 24.2 cm; 27 cm2 Copyright by Holt, Rinehart and Winston.10Holt GeometryAll rights reserved.LESSON1-5Review for MasteryUsing Formulas in Geometry continuedCirclesCircumference AreaModelsDRWords pi times the diameter or 2 times pi times the radiuspi times the square of the radius Formulas C d or C 2r A r2MC 2rA r2C 2(4)A (4)2C 8A 16C 25.1 mA 50.3 m2Find the circumference and area of each circle. Use the key on your calculator. Round to the nearest tenth.7.circle with a radius of 11 inches8.circle with a diameter of 15 millimeters69.1 in.; 380.1 in247.1 mm; 176.7 mm29.IN10.CM56.5 in.; 254.5 in29.4 cm; 7.1 cm2 11.M12. MM81.7 m; 530.9 m2103.7 mm; 855.3 mm2distance around the circlespaceinside the circleCopyright by Holt, Rinehart and Winston.11Holt GeometryAll rights reserved.LESSON1-6Review for MasteryMidpoint and Distance in the Coordinate PlaneThe midpoint of a line segment separates the segment into two halves. You can use the Midpoint Formula to find the midpoint of the segment with endpoints G(1, 2) and H(7, 6). 7XY70-(4, 4)'(1, 2)((7, 6)M x1 x2 ______ 2 ,y1 y2 ______ 2
M 1 7 _____ 2 ,2 6 _____ 2 = M 8 __ 2 ,8 __ 2 = M(4, 4)Find the coordinates of the midpoint of each segment.1. 6XY06 3"(4, 5) ! (2, 5)2. 3XY03 334 (1, 4)3(3, 2) (1, 5) (1, 1)3. _QR with endpoints Q(0, 5) and R(6, 7) (3, 6)4. _JK with endpoints J(1, 4) and K(9, 3) (5, 0.5)Suppose M(3, 1) is the midpoint of_CD and C has coordinates (1, 4). You can use the Midpoint Formula to find the coordinates of D.M (3, 1) M x1 x2 ______ 2,y1 y2 ______ 2
x-coordinate of Dy-coordinate of D3 x1 x2 ______ 2 Set the coordinates equal.1 y1 y2 ______ 2 3 1 x2 ______ 2Replace (x1, y1) with (1, 4).1 4 y2 ______ 2 6 1 x2Multiply both sides by 2.2 4 y25 x2Subtract to solve for x2 and y2.6 y2The coordinates of D are (5, 6).5. M(3, 2) is the midpoint of_RS , and R has coordinates (6, 0). What are the coordinates of S? (12, 4)6. M(7, 1) is the midpoint of_WX , and X has coordinates (1, 5). What are the coordinates of W? (15, 3)M is the midpoint of_HG .Copyright by Holt, Rinehart and Winston.12Holt GeometryAll rights reserved.LESSON1-6Review for MasteryMidpoint and Distance in the Coordinate Plane continuedThe Distance Formula can be used to find the distance d7XY70! (1, 2)"(7, 6)Dbetween points A and B in the coordinate plane.d (x2 x1)2 (y2 y1)2 (7 1 )2 (6 2)2(x1, y1) (1, 2); (x2, y2) (7, 6)62 42Subtract. 36 16Square 6 and 4. 52Add. 7.2Use a calculator.Use the Distance Formula to find the length of each segment or the distance between each pair of points. Round to the nearest tenth.7. _QR with endpoints Q(2, 4) and R(3, 9)8. _EF with endpoints E(8, 1) and F(1, 1)7.1 units 9 units9. T(8, 3) and U(5, 5)10. N(4, 2) and P(7, 1)8.5 units 11.4 unitsYou can also use the Pythagorean Theorem to find distances in the coordinate plane. Find the distance between J and K.c2 a2 b2Pythagorean Theorem
XY
* + CBA 52 62a 5 units and b 6 units 25 36Square 5 and 6. 61Add.c 61or about 7.8Take the square root.Use the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. 11.6XY06 3:(4, 5)9 (0, 1)12.XY
-, 5.7 units 9.4 unitsThe distance d between points A and B is the length of_AB .Side b is 6 units.Side a is 5 units.Copyright by Holt, Rinehart and Winston.4Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.13Holt GeometryAll rights reserved.LESSONReview for MasteryTransformations in the Coordinate Plane 1-7In a transformation, each point of a figure is moved to a new position.Reflection Rotation Translation" #!!" #ABC ABC *+,+,*JKL JKL23 423 4RST RSTA figure is flipped over a line.A figure is turned around a fixed point.A figure is slid to a new position without turning.Identify each transformation. Then use arrow notation to describe the transformation.1. ( '&' (&2. .-0.0- translation; possible answer: FGH FGH reflection; possible answer: MNP MNP3. 7 98987 4. ! "$ #" #! $ reflection; possible answer: WXY WXY rotation; possible answer: ABCD ABCDCopyright by Holt, Rinehart and Winston.14Holt GeometryAll rights reserved.LESSON1-7Review for MasteryTransformations in the Coordinate Plane continuedTriangle QRS has vertices at Q(4, 1), R(3, 4),XY
2 123 31and S(0, 0). After a transformation, the image of the figure has vertices at Q(1, 4), R(4, 3), and S(0, 0). The transformation is a rotation.A translation can be described using a rule such as (x, y) (x 4, y 1).Preimage Apply Rule ImageR(3, 5) R(3 4, 5 1) R(7, 4)S(0, 1) S(0 4, 1 1) S(4, 0)T(2, 1) T(2 4, 1 1) T(6, 2)Draw each figure and its image. Then identify the transformation.5.Triangle HJK has vertices at H(3, 1),
XY *(++*(J(3, 4), and K(0, 0). After a transformation, the image of the figure has vertices at H(1, 3), J(1, 2), and K(4, 2). translation6.Triangle CDE has vertices at C(4, 6),
XY
$ #% %# $D(1, 6), and E(2, 1). After a transformation, the image of the figure has vertices at C(4, 6), D(1, 6), and E(2, 1). reflectionFind the coordinates for each image after the given translation.7.preimage: XYZ at X(6, 1), Y(4, 0), Z(1, 3)rule: (x, y) (x 2, y 5) X(4, 6), Y(6, 5), Z(3, 8)8.preimage: FGH at F(9, 8), G(6, 1), H(2, 4)rule: (x, y) (x 3, y 1) F(6, 9), G(9, 2), H(5, 5)9.preimage: BCD at B(0, 2), C(7, 1), D(1, 5)rule: (x, y) (x 7, y 1) B(7, 1), C(0, 0), D(8, 4)Copyright by Holt, Rinehart and Winston.15Holt GeometryAll rights reserved.LESSONReview for MasteryUsing Inductive Reasoning to Make Conjectures 2-1When you make a general rule or conclusion based on a pattern, you are using inductive reasoning. A conclusion based on a pattern is called a conjecture.Pattern Conjecture Next Two Items8, 3, 2, 7, . . . Each term is 5 more than the previous term.7 5 1212 5 1745The measure of each angle is half the measure of the previous angle.22.511.25Find the next item in each pattern.1. 1 __ 4 ,1 __ 2 ,3 __ 4 , 1, . . .2.100, 81, 64, 49, . . . 1 1 __ 4 363. 3 6 104.
Complete each conjecture. 5.If the side length of a square is doubled, the perimeter of the square is doubled.6.The number of nonoverlapping angles formed by n lines intersecting in a point is 2n.Use the figure to complete the conjecture in Exercise 7.7.The perimeter of a figure that has n of these triangles1 1111031111110411105111106is n 2.Copyright by Holt, Rinehart and Winston.16Holt GeometryAll rights reserved.LESSONReview for MasteryUsing Inductive Reasoning to Make Conjectures continued 2-1Since a conjecture is an educated guess, it may be true or false. It takes only one example, or counterexample, to prove that a conjecture is false.Conjecture: For any integer n, n 4n.n n 4n True or False?3 3 4(3)3 12true0 0 4(0)0 0true2 2 4(2)2 8falsen 2 is a counterexample, so the conjecture is false.Show that each conjecture is false by finding a counterexample.8.If three lines lie in the same plane, then they intersect in at least one point. Possible answer: If the lines are parallel, then they do not intersect.9.Points A, G, and N are collinear. If AG 7 inches and GN 5 inches, then AN 12 inches. Possible answer: If point N is between points A and G, then AN 2 inches. 10.For any real numbers x and y, if x y, then x2 y2. Sample answer: If x 0 and y 1, then x2 y2. 11.The total number of angles in the figure is 3. ! #%$" Sample answer: ABD, DBE, EBC, ABE, DBC 12.If two angles are acute, then the sum of their measures equals the measure of an obtuse angle. Sample answer: m1 25, m2 20Determine whether each conjecture is true. If not, write or draw a counterexample. 13.Points Q and R are collinear.14.If J is between H and K, then HJ JK. ( + * true Copyright by Holt, Rinehart and Winston.5Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.17Holt GeometryAll rights reserved.LESSONA conditional statement is a statement that can be written as an if-then statement, if p, then q.If you buy this cell phone, then you will receive 10 free ringtone downloads.Sometimes it is necessary to rewrite a conditional statement so that it is in if-then form.Conditional:A person who practices putting will improve her golf game.If-Then Form:If a person practices putting, then she will improve her golf game.A conditional statement has a false truth value only if the hypothesis (H) is true and the conclusion (C) is false.For each conditional, underline the hypothesis and double-underline the conclusion.1.If x is an even number, then x is divisible by 2.2. The circumference of a circle is 5 inches if the diameter of the circle is 5 inches.3.If a line containing the points J, K, and L lies in plane P, then J, K, and L are coplanar.For Exercises 46, write a conditional statement from each given statement.4.Congruent segments have equal measures. If segments are congruent, then they have equal measures.5.On Tuesday, play practice is at 6:00.If it is Tuesday, then play practice is at 6:00.6.Adjacent AnglesLinear PairIf two angles form a linear pair, then they are adjacent angles.Determine whether the following conditional is true. If false, give a counterexample.7.If two angles are supplementary, then they form a linear pair.False; two supplementary angles need not be adjacent.The hypothesis comes after the word if.The conclusion comes after the word then.Review for MasteryConditional Statements 2-2Copyright by Holt, Rinehart and Winston.18Holt GeometryAll rights reserved.LESSONC HReview for MasteryConditional Statements continued 2-2The negation of a statement, not p, has the opposite truth value of the original statement.If p is true, then not p is false.If p is false, then not p is true.Statement ExampleTruth ValueConditionalIf a figure is a square, then it has four right angles.TrueConverse:Switch H and C.If a figure has four right angles, then it is a square. FalseInverse:Negate H and C.If a figure is not a square, then it does not have four right angles.FalseContrapositive:Switch and negate H and C.If a figure does not have four right angles, then it is not a square.TrueWrite the converse, inverse, and contrapositive of each conditional statement. Find the truth value of each.8.If an animal is an armadillo, then it is nocturnal.Conv.: If an animal is nocturnal, then it is an armadillo; false. Inv.: If an animal is not an armadillo, then it is not nocturnal; false. Contra.: If an animal is not nocturnal, then it is not an armadillo; true. 9.If y 1, then y2 1. Conv.: If y2 1, then y 1; false. Inv.: If y1, then y21; false. Contra.: If y21, then y1; true. 10.If an angle has a measure less than 90, then it is acute.Conv.: If an angle is acute, then it has a measure less than 90; true. Inv.: If an angle does not have a measure less than 90, then it is not acute; true. Contra.: If an angle is not acute, then it does not have a measure less than 90; true.Copyright by Holt, Rinehart and Winston.19Holt GeometryAll rights reserved.LESSON!"
Review for MasteryUsing Deductive Reasoning to Verify ConjecturesWith inductive reasoning, you use examples to make a conjecture. With deductive reasoning, you use facts, definitions, and properties to draw conclusions and prove that conjectures are true.Given: If two points lie in a plane, then the line containing those points also lies in the plane. A and B lie in plane N.Conjecture: __ AB lies in plane N.One valid form of deductive reasoning that lets you draw conclusions from true facts is called the Law of Detachment.Given If you have $2, then you can buy a snack. You have $2. If you have $2, then you can buy a snack. You can buy a snack. Conjecture You can buy a snack. You have $2.Valid Conjecture? Yes; the conditional is true and the hypothesis is true. No; the hypothesis may or may not be true. For example, if you borrowed money, you could also buy a snack.Tell whether each conclusion uses inductive or deductive reasoning.1.A sign in the cafeteria says that a car wash is being held on the last Saturday of May. Tomorrow is the last Saturday of May, so Justin concludes that the car wash is tomorrow. deductive2.So far, at the beginning of every Latin class, the teacher has had students review vocabulary. Latin class is about to start, and Jamilla assumes that they will first review vocabulary. inductive3.Opposite rays are two rays that have a common endpoint and form a line. __ YX and __ YZ are opposite rays. deductive8 9 :Determine whether each conjecture is valid by the Law of Detachment.4.Given: If you ride the Titan roller coaster in Arlington, Texas, then you will drop 255 feet.Michael rode the Titan roller coaster.Conjecture: Michael dropped 255 feet. valid5.Given: A segment that is a diameter of a circle has endpoints on the circle. _GH has endpoints on a circle.Conjecture:_GH is a diameter. invalid2-3Copyright by Holt, Rinehart and Winston.20Holt GeometryAll rights reserved.LESSONReview for MasteryUsing Deductive Reasoning to Verify Conjectures continuedAnother valid form of deductive reasoning is the Law of Syllogism. It is similar to the Transitive Property of Equality.Transitive Property of Equality Law of SyllogismIf y 10x and 10x 20, then y 20.Given: If you have a horse, then you have to feed it. If you have to feed a horse, then you have to get up early every morning.Conjecture: If you have a horse, then you have to get up early every morning.Determine whether each conjecture is valid by the Law of Syllogism.6.Given: If you buy a car, then you can drive to school. If you can drive to school, then you will not ride the bus.Conjecture: If you buy a car, then you will not ride the bus. valid7.Given: If K is obtuse, then it does not have a measure of 90. If an angle does not have a measure of 90, then it is not a right angle.Conjecture: If K is obtuse, then it is not a right angle. valid8.Given: If two segments are congruent, then they have the same measure. If two segments each have a measure of 6.5 centimeters, then they are congruent.Conjecture: If two segments are congruent, then they each have a measure of 6.5 centimeters. invalidDraw a conclusion from the given information.9.If LMN is translated in the coordinate plane, then it has the same size and shape as its preimage. If an image and preimage have the same size and shape, then the figures have equal perimeters. LMN is translated in the coordinate plane.LMN and LMN have equal perimeters. 10.If R and S are complementary to the same angle,2 3then the two angles are congruent. If two angles are congruent, then they are supplementary to the same angle. R and S are complementary to the same angle.R and S are supplementary to the same angle.2-3Copyright by Holt, Rinehart and Winston.6Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.21Holt GeometryAll rights reserved.LESSONp qq pp qReview for MasteryBiconditional Statements and Definitions 2-4A biconditional statement combines a conditional statement, if p, then q, with its converse, if q, then p.Conditional: If the sides of a triangle are congruent, then the angles are congruent.Converse:If the angles of a triangle are congruent, then the sides are congruent.Biconditional: The sides of a triangle are congruent if and only if the angles are congruent.Write the conditional statement and converse within each biconditional.1.Lindsay will take photos for the yearbook if and only if she doesnt play soccer.Conditional: If Lindsay takes photos for the yearbook, then she doesnt play soccer. Converse: If Lindsay doesnt play soccer, then she will take photos for the yearbook.2.mABC mCBD if and only if __ BC is an angle bisector of ABD. !"#$Conditional: If mABC mCBD, then ___ BC is an angle bisector of ABD. Converse: If ___ BC is an angle bisector of ABD, then mABC mCBD.For each conditional, write the converse and a biconditional statement.3.If you can download 6 songs for $5.94, then each song costs $0.99.Converse: If each song costs $0.99, then you can download 6 songs for $5.94.Biconditional: You can download 6 songs for $5.94 if and only if each song 4.If a figure has 10 sides, then it is a decagon.Converse: If a figure is a decagon, then it has 10 sides. Biconditional: A figure has 10 sides if and only if it is a decagon.costs $0.99.Copyright by Holt, Rinehart and Winston.22Holt GeometryAll rights reserved.LESSONReview for MasteryBiconditional Statements and Definitions continued 2-4A biconditional statement is false if either the conditional statement is false or its converse is false.The midpoint of_QR is M(3, 3) if and only if the endpoints are Q(6, 1) and R(0, 5).Conditional: If the midpoint of_QR is M(3, 3), then the0XY3-(3, 3)1(6, 1)2(0, 5)3endpoints are Q(6, 1) and R(0, 5). falseConverse: If the endpoints of_QR are Q(6, 1) and R(0, 5),then the midpoint of_QR is M(3, 3). trueThe conditional is false because the endpoints of_QR could be Q(3, 6) and R(3, 0). So the biconditional statement is false.Definitions can be written as biconditionals.Definition:Circumference is the distance around a circle.Biconditional: A measure is the circumference if and only if it is the distance around a circle.Determine if each biconditional is true. If false, give a counterexample.5.Students perform during halftime at the football games if and only if they are in the high school band. False; possible answer: members of flag corps perform during halftime.6.An angle in a triangle measures 90 if and only if the triangle is a right triangle. true7.a 4 and b 3 if and only if ab 12. False; possible answer: a 2 and b 6Write each definition as a biconditional.8.An isosceles triangle has at least two congruent sides. A triangle is isosceles if and only if it has at least two congruent sides. 9.Deductive reasoning requires the use of facts, definitions, and properties to draw conclusions. You use deductive reasoning if and only if you use facts, definitions, and properties to draw conclusions.Copyright by Holt, Rinehart and Winston.23Holt GeometryAll rights reserved.LESSON x xSubtr. Prop. of 5 xSimplify. x 5Sym. Prop. of y 4 _____ 7(7) 3(7)Mult. Prop. of y 4 21Simplify.4 4Subtr. Prop. of y 17Simplify.4t 12 20Distr. Prop. 12 12Add Prop. of 4t 8Simplify. 4t __ 48 ___ 4Div. Prop. of t 2Simplify.Review for MasteryAlgebraic ProofA proof is a logical argument that shows a conclusion is true. An algebraic proof uses algebraic properties, including the Distributive Property and the properties of equality.Properties of EqualitySymbols ExamplesAdditionIf a b, then a c b c. If x 4, then x 4 4 4.Subtraction If a b, then a c b c. If r 1 7, then r 1 1 7 1.Multiplication If a b, then ac bc. Ifk __ 2 8, thenk __ 2 (2) 8(2). Division If a 2 and c0, thena __ cb __ c . If 6 3t, then6 __ 3 3t __ 3 .Reflexive a a 15 15Symmetric If a b, then b a. If n 2, then 2 n.Transitive If a b and b c, then a c. If y 32 and 32 9, then y 9.Substitution If a b, then b can be substituted for a in any expression.If x 7, then 2x 2(7).When solving an algebraic equation, justify each step by using a definition, property, or piece of given information.2(a 1) 6Given equation2a 2 6Distributive Property 2 2Subtraction Property of Equality2a 8Simplify. 2a ___ 2 8 ___ 2Division Property of Equalitya 4Simplify.Solve each equation. Write a justification for each step.1. n __ 6 3 10Given equation2.5 x 2xGiven equation 3 3Add. Prop. of n __ 6 13Simplify.
n __ 6 (6) 13(6)Mult. Prop. of n 78Simplify.3. y 4 _____ 7 3Given equation4.4(t 3) 20Given equation2-5Copyright by Holt, Rinehart and Winston.24Holt GeometryAll rights reserved.LESSONReview for MasteryAlgebraic Proof continued 2-5When writing algebraic proofs in geometry, you can also use definitions, postulates, properties, and pieces of given information to justify the steps.mJKM mMKLDefinition of congruent angles*-XX,+(5x 12) 4xSubstitution Property of Equalityx 12 0Subtraction Property of Equalityx 12Addition Property of EqualityProperties of CongruenceSymbols ExamplesReflexivefigure A figure A CDE CDESymmetricIf figure A figure B, then figure B figure A.If_JK _LM , then_LM _JK .TransitiveIf figure A figure B and figure B figure C, then figure A figure C.If N P and P Q, then N Q.Write a justification for each step.5.CE CD DE Segment Addition Postulate 3X7 8# % $6X 6x 8 (3x 7) Substitution Property of Equality 6x 15 3x Simplify. 3x 15 Subtraction Property of Equalityx 5 Division Property of Equality6.mPQR mPQS mSQR Angle Addition Postulate XX230190 2x (4x 12) Substitution Property of Equality90 6x 12 Simplify.102 6x Addition Property of Equality17 x Division Property of EqualityIdentify the property that justifies each statement.7.If ABC DEF, then DEF ABC.8.1 2 and 2 3, so 1 3. Symmetric Property of Congruence Transitive Property of Congruence9.If FG HJ, then HJ FG.10. _WX _WX Symmetric Property of Equality Reflexive Property of CongruenceCopyright by Holt, Rinehart and Winston.7Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.25Holt GeometryAll rights reserved.LESSONHypothesisDeductive Reasoning Definitions Properties Postulates TheoremsConclusionReview for MasteryGeometric Proof 2-6To write a geometric proof, start with the hypothesis of a conditional.Apply deductive reasoning.Prove that the conclusion of the conditional is true.Conditional: If __ BD is the angle bisector of ABC, and ABD 1, then DBC 1.Given: __ BD is the angle bisector of ABC, and ABD 1. 1#"$!Prove: DBC 1Proof:1. __ BD is the angle bisector of ABC.1.Given2.ABD DBC2.Def. of bisector3.ABD 13.Given4.DBC 14.Transitive Prop. of 1.Given:N is the midpoint of_MP , Q is themidpoint of_RP , and_PQ _NM . 0 12-.Prove: _PN _QR Write a justification for each step.Proof:1.N is the midpoint of_MP .1. Given2.Q is the midpoint of_RP .2. Given3. _PN _NM3. Def. of midpoint4. _PQ _NM4. Given5. _PN _PQ5. Transitive Prop. of 6. _PQ _QR6. Def. of midpoint7._PN _QR7. Transitive Prop. of Copyright by Holt, Rinehart and Winston.26Holt GeometryAll rights reserved.LESSONReview for MasteryGeometric Proof continued 2-6A theorem is any statement that you can prove. You can use two-column proofs and deductive reasoning to prove theorems.Congruent Supplements TheoremIf two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent.Right Angle Congruence TheoremAll right angles are congruent.Here is a two-column proof of one case of the Congruent Supplements Theorem.Given: 4 and 5 are supplementary and 5 and 6 are supplementary. 465 7Prove:4 6Proof:Statements Reasons1. 4 and 5 are supplementary. 1. Given2. 5 and 6 are supplementary. 2. Given3. m4 m5 180 3. Definition of supplementary angles4. m5 m6 180 4. Definition of supplementary angles5. m4 m5 m5 m6 5. Substitution Property of Equality6. m4 m6 6. Subtraction Property of Equality7. 4 6 7. Definition of congruent anglesFill in the blanks to complete the two-column proof12of the Right Angle Congruence Theorem.2.Given: 1 and 2 are right angles.Prove: 1 2Proof:Statements Reasons1. a. 1 and 2 are right angles. 1. Given2. m1 902. b. Definition of right angle3. c. m2 90 3. Definition of right angle4. m1 m24. d. Transitive Property of Equality5. e. 1 2 5. Definition of congruent anglesCopyright by Holt, Rinehart and Winston.27Holt GeometryAll rights reserved.LESSONReview for MasteryFlowchart and Paragraph Proofs 2-7In addition to the two-column proof, there are other types of proofs that you can use to prove conjectures are true.Flowchart Proof Uses boxes and arrows. Steps go left to right or top to bottom, as shown by arrows. The justification for each step is written below the box.You can write a flowchart proof of the Right Angle Congruence Theorem.Given:1 and 2 are right angles.12Prove:1 21 and 2 are rt. .Givenm190, m290Def. of rt. m1m2Trans. Prop. of 12Def. of1.Use the given two-column proof to write a flowchart proof.Given:V is the midpoint of_SW , and W is the midpoint of_VT .3674Prove: _SV _WT Two-Column Proof:Statements Reasons1. V is the midpoint of_SW . 1. Given2. W is the midpoint of_VT . 2. Given3._SV _VW ,_VW _WT3. Definition of midpoint4._SV _WT4. Transitive Property of Equality6ISTHEMIDPOINTOF37'IVEN3667$EFOFMIDPOINT7ISTHEMIDPOINTOF64'IVEN6774$EFOFMIDPOINT36744RANS0ROPOFCopyright by Holt, Rinehart and Winston.28Holt GeometryAll rights reserved.LESSONReview for MasteryFlowchart and Paragraph Proofs continued 2-7To write a paragraph proof, use sentences to write a paragraph that presents the statements and reasons.You can use the given two-column proof to write a paragraph proof.Given: _AB _BC and_BC _DE"$%!#Prove: _AB _DE Two-Column Proof:Statements Reasons1._AB _BC ,_BC _DE1. Given2. AB BC, BC DE 2. Definition of congruent segments3. AB DE 3. Transitive Property of Equality4._AB _DE4. Definition of congruent segmentsParagraph Proof: It is given that_AB _BC and_BC _DE , so AB BC and BC DE by the definition of congruent segments. By the Transitive Property of Equality, AB DE. Thus, by the definition of congruent segments,_AB _DE .2.Use the given two-column proof to write a paragraph proof.Given:JKL is a right angle. 12*+,Prove:1 and 2 are complementary angles.Two-Column Proof:Statements Reasons1. JKL is a right angle. 1. Given2. mJKL 90 2. Definition of right angle3. mJKL m1 m2 3. Angle Addition Postulate4. 90 m1 m2 4. Substitution5. 1 and 2 are complementary angles. 5. Definition of complementary anglesParagraph Proof: Since JKL is a right angle, mJKL 90by the definition of right angle. By the Angle Addition Postulate, mJKL m1 m2. Using substitution, 90 m1 m2. Thus, by the definition of complementary angles, 1 and 2 are complementary angles.Copyright by Holt, Rinehart and Winston.8Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.29Holt GeometryAll rights reserved.LESSONReview for MasteryLines and Angles 3-1Lines Description Examplesparallellines that lie in the same plane and do not intersectsymbol: ||MK
perpendicularlines that form 90 anglessymbol: skew lines that do not lie in the same plane and do not intersectParallel planes are planes that do not intersect. For example, the top and bottom of a cube represent parallel planes.Use the figure for Exercises 13. Identify each of the following.1.a pair of parallel lines JGH
g h2.a pair of skew linesj and h3.a pair of perpendicular linesj gUse the figure f or Exercises 4 9. Identify each of the following. $%&'(*4.a segment that is parallel to_DG5.a segment that is perpendicular to_GH Possible answers:_EHor_FJ Sample answer:_HJ 6.a segment that is skew to_JF7.one pair of parallel planesSample answer:_DEplane DEF plane GHJ8.one pair of perpendicular segments, 9.one pair of skew segments, not including_GH not including_JF Sample answer:_DE _EFSample answer:_HEand_DF mk k and m are skew.Copyright by Holt, Rinehart and Winston.30Holt GeometryAll rights reserved.LESSONReview for MasteryLines and Angles continued 3-1A transversal is a line that intersects two lines in a plane at different points. Eight angles are formed. Line t is a transversal of lines a and b. TA1 234567 8BAngle Pairs Formed by a TransversalAngles Description Examplescorrespondingangles that lie on the same side of the transversal and on the same sides of the other two linesTA 48Balternate interiorangles that lie on opposite sides of the transversal, between the other two linesTA 45Balternate exteriorangles that lie on opposite sides of the transversal, outside the other two linesTA27Bsame-side interiorangles that lie on the same side of the transversal, between the other two lines; also called consecutive interior anglesTA 46BUse the figure for Exercises 1013. Give an example of each type of angle pair. 1 2 345 6 7 8 10.corresponding angles11.alternate exterior angles Sample answer: 1 and 3 Sample answer: 1 and 8 12.same-side interior angles 13.alternate interior angles Sample answer: 2 and 3 Sample answer: 2 and 7Use the figure for Exercises 1416. Identify the transversal and classify each angle pair. 1234MNP14.1 and 2 transv. n; same-side int. 15.2 and 416.3 and 4 transv. m; alt. ext. transv. p; corr. Copyright by Holt, Rinehart and Winston.31Holt GeometryAll rights reserved.LESSON3-2Review for MasteryAngles Formed by Parallel Lines and TransversalsAccording to the Corresponding Angles Postulate, if two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.R
STDetermine whether each pair of angles is congruent according to the Corresponding Angles Postulate.12 341. 1 and 22. 3 and 4no yesFind each angle measure.167 142+*(X3.m14.mHJK67 142X X!#" XX-0 1.,5.mABC6.mMPQ92 125 1 3 2 4Copyright by Holt, Rinehart and Winston.32Holt GeometryAll rights reserved.LESSON3-2Review for MasteryAngles Formed by Parallel Lines and Transversals continuedIf two parallel lines are cut by a transversal, then the following pairs of angles are also congruent.Angle Pairs Hypothesis Conclusionalternate interior angles2 63 7CTD2 36 7alternate exterior angles14 85QTR1 45 8If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary. Find each angle measure. 3111 47.m38.m4 111 90 138X 234AA-0.9.mRST10.mMNP 138 56 Y Y7:8 N N!"#$ 11.mWXZ12.mABC 130 118m5 m6 180 m1 m2 180Copyright by Holt, Rinehart and Winston.9Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.33Holt GeometryAll rights reserved.LESSONReview for MasteryProving Lines Parallel 3-3Converse of the Corresponding Angles Postulate If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel.You can use the Converse of the Corresponding Angles Postulate to show that two lines are parallel. 1 2QR3 4Given: 1 31 31 3 are corresponding angles.q || rConverse of the Corresponding Angles PostulateGiven: m2 3x, m4 (x 50), x 25m2 3(25) 75Substitute 25 for x.m4 (25 50) 75Substitute 25 for x.m2 m4Transitive Property of Equality2 4Definition of congruent anglesq || rConverse of the Corresponding Angles PostulateFor Exercises 1 and 2, use the Converse of the Corresponding Angles Postulate and the given information to show that c || d.1.Given: 2 42 42 and 4 are corr. .c || dConv. of Corr. Post.2.Given: m1 2x, m3 (3x 31), x 31m1 2x 2(31) 62Substitute 31 for x.m3 (3x 31) 3(31) 31 62Substitute 31 for x.m1 m3Trans. Prop. of 1 3Def. of c || dConv. of Corr. Post.12DC34Copyright by Holt, Rinehart and Winston.34Holt GeometryAll rights reserved.LESSON3-3Review for MasteryProving Lines Parallel continuedYou can also prove that two lines are parallel by using the converse of any of the other theorems that you learned in Lesson 3-2.Theorem Hypothesis ConclusionConverse of the Alternate Interior Angles Theorem2 ATB32 3a || bConverse of the Alternate Exterior Angles Theorem4FTG11 4f || gConverse of the Same-Side Interior Angles Theorem1 ST2m1 m2 180s || tFor Exercises 35, use the theorems and the given information to show that j k.3.Given: 4 54 54 and 5 are alt. int. .j kConv. of Alt. Int. Thm.4.Given: m3 12x, m5 18x, x 6m3 12(6) 72Substitute 6 for x.m5 18(6) 108Substitute 6 for x.m3 m5 72 108 180Add angle measures.j kConv. of Same-Side Int. Thm.5.Given: m2 8x, m7 (7x 9), x 9m2 8(9) 72Substitute 9 for x.m7 7(9) 9 72Substitute 9 for x.m2 m7Trans. Prop. of 2 7Def. of j kConv. of Alt. Ext. Thm.JK1 23 45 67 8Copyright by Holt, Rinehart and Winston.35Holt GeometryAll rights reserved.LESSON3-4Review for MasteryPerpendicular LinesThe perpendicular bisector of a segment is a line perpendicular to the segment at the segments midpoint.B2 3The distance from a point to a line is the length of the shortest segment from the point to the line. It is the length of the perpendicular segment that joins them.3 47
X5You can write and solve an inequality for x.WU WT _WT is the shortest segment.x 1 8Substitute x 1 for WU and 8 for WT. 1 1Subtract 1 from both sides of the equality.x 7Use the figure for Exercises 1 and 2.1.Name the shortest segment from point K to __ LN.
_KM 2.Write and solve an inequality for x. , -+
X. x 5 14; x 9Use the figure for Exercises 3 and 4.3.Name the shortest segment from point Q to ___ GH.
_QH 4.Write and solve an inequality for x. x 2 9; x 11 '( 1
XLine b is the perpendicular bisector of_RS .The shortest segment from W to __ SU is_WT.Copyright by Holt, Rinehart and Winston.36Holt GeometryAll rights reserved.LESSONReview for MasteryPerpendicular Lines continued 3-4You can use the following theorems about perpendicular lines in your proofs.Theorem ExampleIf two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular.Symbols:2 intersecting lines form lin. pair of lines .AB1 21 and 2 form a linear pair and 1 2, so a b.Perpendicular Transversal TheoremIn a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line.Symbols: Transv. Thm.DCHh c and c d, so h d.If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other.Symbols:2 lines to same line 2 lines .KJ
j and k , so j k.5.Complete the two-column proof.Given:1 2, s tProve:r tProof: Statements Reasons1. 1 2 1. Given2. a. r s 2. Conv. of Alt. Int. Thm.3. s t3. b. Given4. r t4. c. Transv. Thm.TS12RCopyright by Holt, Rinehart and Winston.10Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.37Holt GeometryAll rights reserved.LESSON3-5Review for MasterySlopes of LinesThe slope of a line describes how steep the line is. You can find the slope by writing the ratio of the rise to the run.slope rise ____ run3 __ 6 1 __ 2 You can use a formula to calculate the slope m of the line through points (x1, y1) and (x2, y2).m =rise ____ run=y2 y1 ______ x2 x1 To find the slope of __ AB using the formula, substitute (1, 3) for (x1, y1) and (7, 6) for (x2, y2).Use the slope formula to determine the slope of each line. 0XY222(*2 0XY2# $21. __ HJ2. ___ CD 2 __ 3 0 0XY2,-2 3 0XY223223. __ LM4. __ RS 2 4 __ 3 Change in x-values0XY4"(7, 6)!(1, 3)4rise: go up 3 unitsrun: go right 6 unitsm = y2 y1 ______ x2 x1Slope formula=6 3 _____ 7 1Substitution=3 __ 6Simplify.=1 __ 2Simplify.Change in y-valuesCopyright by Holt, Rinehart and Winston.38Holt GeometryAll rights reserved.LESSONReview for MasterySlopes of Lines continued 3-5Slopes of Parallel and Perpendicular Lines0XY2,-.0242slope of __ LM = 3slope of __ NP = 3Parallel lines have the same slope.0XY2.012224slope of __ NP 3slope of ___ QR 1 __ 3 product of slopes:3 1 __ 3 1Perpendicular lines have slopes that are opposite reciprocals. The product of the slopes is 1.Use slopes to determine whether each pair of distinct lines is parallel, perpendicular, or neither.5.slope of ___ PQ 56.slope of __ EF 3 __ 4 slope of __ JK 1 __ 5 slope of __ CD 3 __ 4 perpendicular parallel7.slope of ___ BC 5 __ 38.slope of __ WX 1 __ 2 slope of __ ST 3 __ 5slope of __ YZ 1 __ 2 perpendicular neitherGraph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither.
X'(* &Y
X2 345Y
9. __ FG and __ HJ for F(1, 2), G(3, 4), 10. __ RS and __ TU for R(2, 3), S(3, 3), H(2, 3), and J(4, 1) T(3, 1), and U(3, 1) perpendicular neitherCopyright by Holt, Rinehart and Winston.39Holt GeometryAll rights reserved.LESSONslope y-intercept slopeReview for MasteryLines in the Coordinate Plane 3-6Slope-Intercept Form Point-Slope Formy mx by 4x 7y y1 m(x x1)point on the line:y 2 1 __ 3 (x 5)(x1, y1) (5, 2)Write the equation of the line through (0, 1) and (2, 7) in slope-intercept form.Step 1: Find the slope.m y2 y1 ______x2 x1 Formula for slope7 1 _____ 2 0 6 __ 2 3Step 2: Find the y-intercept.y mx bSlope-intercept form1 3(0) bSubstitute 3 for m, 0 for x, and 1 for y.1 bSimplify.Step 3: Write the equation.y mx bSlope-intercept formy 3x 1Substitute 3 for m and 1 for b.Write the equation of each line in the given form.1.the line through (4, 2) and (8, 5) in 2.the line through (4, 6) with slope1 __ 2 slope-intercept formin point-slope formy 3 __ 4 x 1 y 6 1 __ 2 (x 4)3.the line through (5, 1) with slope 2 4.the line with x-intercept 5 and in point-slope formy-intercept 3 in slope-intercept formy 1 2(x 5)y 3 __ 5 x 35.the line through (8, 0) with slope 3 __ 4 6.the line through (1, 7) and (6, 7) in slope-intercept formin point-slope formy 3 __ 4 x 6y 7 0Copyright by Holt, Rinehart and Winston.40Holt GeometryAll rights reserved.LESSON3-6Review for MasteryLines in the Coordinate Plane continuedYou can graph a line from its equation.Consider the equation y 2 __ 3 x 2.y-intercept 2 slope 2 __ 3
XY
First plot the y-intercept (0, 2). Use rise 2 and run 3 to find another point. Draw the line containing the two points.Parallel Lines Intersecting Lines Coinciding Lines
XY
same slopedifferent y-intercepts XY
different slopes XY
same slopesame y-interceptGraph each line.
X
Y
X
Y
XY
7. y x 28. y 1 __ 3 x 39. y 2 1 __ 4 (x 1)Determine whether the lines are parallel, intersect, or coincide. 10. y 2x 511. y 1 __ 3 x 4y 2x 1x 3y 12parallel coincide 12. y 5x 213.5y 2x 1x 4y 8y 2 __ 5 x 3intersect parallelrun: go left 3 unitsrise: go up 2 unitsy 1 __ 3 x 2y 1 __ 3 xy 1 __ 2 x 2y 2x 1y 2 __ 3 x 12x 3y 3Copyright by Holt, Rinehart and Winston.11Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.41Holt GeometryAll rights reserved.LESSONReview for MasteryClassifying Triangles 4-1You can classify triangles by their angle measures. An equiangular triangle, for example, is a triangle with three congruent angles.Examples of three other triangle classifications are shown in the table.Acute Triangle Right Triangle Obtuse Triangleall acute angles one right angle one obtuse angleYou can use angle measures to classify JML at right.JLM and JLK form a linear pair, so they are supplementary.mJLM mJLK 180Def. of supp. mJLM 120 180SubstitutionmJLM 60Subtract.Since all the angles in JLM are congruent, JLM is an equiangular triangle.Classify each triangle by its angle measures.1.
2.
3.
right obtuse acuteUse the figure to classify each triangle by its angle measures.4. DFGright5. DEGacute6. EFG obtuse!"#N!"# is equiangular.!# "6060 60
*- +,
$' &%
JKL is obtuse so JLK is an obtuse triangle.Copyright by Holt, Rinehart and Winston.42Holt GeometryAll rights reserved.LESSONReview for MasteryClassifying Triangles continued 4-1You can also classify triangles by their side lengths.Equilateral Triangle Isosceles Triangle Scalene Triangleall sides congruentat least two sides congruent no sides congruentYou can use triangle classification to find the side lengths of a triangle.Step 1Find the value of x.QR RSDef. of segs.4x 3x 5Substitutionx 5Simplify.Step 2Use substitution to find the length of a side.4x 4(5)Substitute 5 for x. 20Simplify.Each side length of QRS is 20.Classify each triangle by its side lengths.7.EGF isosceles8.DEF scalene9.DFG isoscelesFind the side lengths of each triangle. 10. XX11. XXX 9; 9; 9 7; 7; 4
X X21 3$' &%
Copyright by Holt, Rinehart and Winston.43Holt GeometryAll rights reserved.LESSONReview for MasteryAngle Relationships in Triangles 4-2mC 90 39 51According to the Triangle Sum Theorem, the sum of the angle
*, +measures of a triangle is 180. mJ mK mL 62 73 45 180The corollary below follows directly from the Triangle Sum Theorem.Corollary ExampleThe acute angles of a right triangle are complementary.
% $#mC mE 90Use the figure for Exercises 1 and 2.1.Find mABC.
!$#"472.Find mCAD.38Use RST for Exercises 3 and 4.3.What is the value of x? (7X13)(4X9)(2X2)24 3144.What is the measure of each angle?mR 85; mS 30; mT 65What is the measure of each angle?
. ,- "#! X5765. L6. C7. W49 39.8 (90 x)Copyright by Holt, Rinehart and Winston.44Holt GeometryAll rights reserved.LESSONAn exterior angle of a triangle is formed by
one side of the triangle and the extension of an adjacent side.1 and 2 are the remote interior angles of 4 because they are not adjacent to 4.Exterior Angle TheoremThe measure of an exterior angle of a
triangle is equal to the sum of the measures of its remote interior angles.Third Angles TheoremIf two angles of one triangle are congruent
to two angles of another triangle, then the third pair of angles are congruent.Find each angle measure.
(&'* 68(4X5)3X! "#$8.mG9.mD 51 41Find each angle measure. (6X10)(7X2),+. 01- XX354 2 10.mM and mQ11.mT and mR 82; 82 33; 334-2Review for MasteryAngle Relationships in Triangles continuedremoteinterior anglesexterior anglem4 m1 m2Copyright by Holt, Rinehart and Winston.12Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.45Holt GeometryAll rights reserved.LESSONTriangles are congruent if they have the same size and shape. Their corresponding parts, the angles and sides that are in the same positions, are congruent.!# " +N!"#N*+,,* Corresponding PartsCongruent Angles Congruent SidesA JB KC L _AB _JK _BC _KL _CA _LJ To identify corresponding parts of congruent triangles, look at the order of the vertices in the congruence statement such as ABC JKL.Given: XYZ NPQ. Identify the congruent corresponding parts.918.:01.Z Q2. _YZ _PQ 3.P Y4.X N5. _NQ _XZ 6. _PN _YX Given: EFG RST. Find each value below.% '234&(4X6)(5Y2)3Z8 Z4287.x 218.y 69.mF 6210.ST 104-3Review for MasteryCongruent TrianglesCopyright by Holt, Rinehart and Winston.46Holt GeometryAll rights reserved.LESSONYou can prove triangles congruent by using the definition of congruence.Given: D and B are right angles. $ "#% !DCE BCAC is the midpoint of_DB . _ED _AB ,_EC _AC Prove: EDC ABCProof:Statements Reasons1. D and B are rt. . 1. Given2. D B 2. Rt. Thm.3. DCE BCA 3. Given4. E A 4. Third Thm.5. C is the midpoint of_DB . 5. Given6._DC _BC6. Def. of mdpt.7._ED _AB ,_EC _AC7. Given8. EDC ABC 8. Def. of s 11.Complete the proof.Given: Q R .2013P is the midpoint of_QR .
_NQ _SR,_NP _SP Prove: NPQ SPRProof:Statements Reasons1. Q R 1. Given2. NPQ SPR 2. a. Vert. Thm.3. N S 3. b. Third Thm.4. P is the midpoint of_QR . 4. c. Given5. d. _QP _RP 5. Def. of mdpt.6._NQ _SR ,_NP _SP6. e. Given7. NPQ SPR 7. f. Def. of sReview for MasteryCongruent Triangles continued 4-3Copyright by Holt, Rinehart and Winston.47Holt GeometryAll rights reserved.LESSONSide-Side-Side (SSS) Congruence PostulateIf three sides of one triangle are congruent to three sides05 4231of another triangle, then the triangles are congruent. _QR _TU ,_RP _US , and_PQ _ST , so PQR STU.You can use SSS to explain why FJH FGH. (&* 'It is given that_FJ _FG and that_JH _GH . By the Reflex. Prop. of ,_FH _FH . So FJH FGH by SSS.Side-Angle-Side (SAS) Congruence PostulateIf two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.( ,+ . - *N(*+N,-.Use SSS to explain why the triangles in each pair are congruent.,-*+ $#"!1. JKM LKM2. ABC CDAIt is given that_JK _LK and that It is given that_AB _CD and that_JM _LM . By the Reflex. Prop._AD _CB . By the Reflex. Prop.of ,_KM _KM . So JKM of ,_AC _AC. So ABC LKM by SSS. CDA by SSS.3.Use SAS to explain why WXY WZY.79: 8It is given that_ZW _XW and thatZWY XWY. By the Reflex.Prop. of ,_WY _WY. SoWXY WZY by SAS.4-4Review for MasteryTriangle Congruence: SSS and SASK is the included angle of_HK and_KJ.N is the included angle of_LN and_NM.Copyright by Holt, Rinehart and Winston.48Holt GeometryAll rights reserved.LESSONYou can show that two triangles are congruent by using SSS and SAS.Show that JKL FGH for y 7. HG y 6mG 5y 5FG 4y 1 7 6 13 5(7) 5 40 4(7) 1 27 HG LK 13, so_HG _LK by def. of segs. mG = 40, so G K by def. of . FG JK 27, so_FG _JK by def. of segs. Therefore JKL FGH by SAS.Show that the triangles are congruent for the given value of the variable. "$ #'(&X2X2X396 8 6871203N87N1721(36N5)1134.BCD FGH, x 65.PQR VWX, n 3 BD FH 6, so_BD _FH by PR VX 17, so_PR _VX by def. of segs. BC FG 8, def. of segs. mP so_BC _FG by def. of segs. mV 113, so P V by CD GH 9, so_CD _GH def. of . PQ VW 21, so by def. of segs. Therefore _PQ _VW by def. of segs. BCD FGH by SSS. Therefore PQR VWX by SAS.6.Complete the proof.Given: T is the midpoint of_VS . 6 324 _RT _VS Prove: RST RVTStatements Reasons1. T is the midpoint of_VS . 1. Given2. a. _VT _ST 2. Def. of mdpt.3._RT _VS3. b. Given4. RTV and RTS are rt. 4. c. Def. of lines5. d. RTV RTS5. Rt. Thm.6._RT _RT6. e. Reflex. Prop. of 7. RST RVT 7. f. SAS 4-4Review for MasteryTriangle Congruence: SSS and SAS continued* +' &(,27134Y1(5Y5)Y640Copyright by Holt, Rinehart and Winston.13Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.49Holt GeometryAll rights reserved.LESSONAngle-Side-Angle (ASA) Congruence PostulateIf two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.! $# & % "N!"#N$%&Determine whether you can use ASA to prove the triangles congruent. Explain.,+. 01CMCM-8:M9%'M&1. KLM and NPQ2. EFG and XYZYes; K N,_KL _NP, and No; you need to know thatL P as given._GF _ZY.. ,-0 +756433. KLM and PNM, given that M is the 4. STW and UTVmidpoint of_NL No; you need to know thatYes; W V and_TW _TV asNMP LMK. given. STW UTV by the Vert. Thm.Review for MasteryTriangle Congruence: ASA, AAS, and HL 4-5 _AC is the included side of A and C. _DF is the included side of D and F.Copyright by Holt, Rinehart and Winston.50Holt GeometryAll rights reserved.LESSONAngle-Angle-Side (AAS) Congruence TheoremIf two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent.* &, + ' (N&'(N*+,Special theorems can be used to prove right triangles congruent.Hypotenuse-Leg (HL) Congruence TheoremIf the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.* + ., 0-N*+,N-.05.Describe the corresponding parts and the justifications $ #"!for using them to prove the triangles congruent by AAS.Given:_BD is the angle bisector of ADC.Prove: ABD CBD A C (Given), ADB CDB (Def. of bisector), _BD _BD (Reflex. Prop. of )Determine whether you can use the HL Congruence Theorem to prove the triangles congruent. If yes, explain. If not, tell what else you need to know. 5876 243 106.UVW WXU7.TSR PQR Yes;_UV _WX (Given) and No; you need to know that _UW _UW (Reflex. Prop. of ) _TR _PR .4-5Review for MasteryTriangle Congruence: ASA, AAS, and HL continued _FH is a nonincluded side of F and G._JL is a nonincluded side of J and K.Copyright by Holt, Rinehart and Winston.51Holt GeometryAll rights reserved.LESSONCorresponding Parts of Congruent Triangles are Congruent (CPCTC) is useful in proofs. If you prove that two triangles are congruent, then you can use CPCTC as a justification for proving corresponding parts congruent.Given:_AD _CD ,_AB _CB Prove: A CProof:!"#"Given SSS! #CPCTCN!"$N#"$"$"$Reflex. Prop of !$#$GivenComplete each proof. 01.-,1.Given:PNQ LNM,_PN _LN , N is the midpoint of_QM .Prove:_PQ _LM Proof:0.,.SAS01,-CPCTCO01.O,-.Givenc.d.Givena.0.1,.-Given.is themdpt. of -1.Def. of midpt.1.-.b.2.Given:UXW and UVW are right s.56 87 _UX _UV Prove: X VProof:Statements Reasons1. UXW and UVW are rt. s. 1. Given2._UX _UV2. a. Given3._UW _UW3. b. Reflex. Prop. of 4. c. UXW UVW4. d. HL5. X V 5. e. CPCTCReview for MasteryTriangle Congruence: CPCTC 4-6" #!$Copyright by Holt, Rinehart and Winston.52Holt GeometryAll rights reserved.LESSONYou can also use CPCTC when triangles are on the coordinate plane.Given:C(2, 2), D(4, 2), E(0, 2),0X#$ %&'(Y222F(0, 1), G(4, 1), H(4, 3)Prove: CED FHGStep 1 Plot the points on a coordinate plane.Step 2Find the lengths of the sides of each triangle. Use the Distance Formula if necessary.d (x2 x1)2 (y2 y1)2 CD (4 2)2 (2 2)2FG (4 0)2 (1 1)2 4 16 2 5 16 4 25 DE 4GH 4EC (2 0)2 [2 (2)]2 HF [0 (4)]2 (1 3)2 4 16 2 5= 16 4 25 So,_CD _FG ,_DE _GH , and_EC _HF . Therefore CDE FGH by SSS, and CED FHG by CPCTC.Use the graph to prove each congruence statement. 0X987312Y2232 0X*+,!#"Y23223.RSQ XYW4.CAB LJK QR WX 13 , AB JK 5, BC KL RS XY 7, SQ YW 34 . 10 , CA LJ 53 . So So QRS WXY by SSS, and ABC JKL by SSS, and RSQ XYW by CPCTC. CAB LJK by CPCTC.5.Use the given set of points to prove PMN VTU.M(2, 4), N(1, 2), P(3, 4), T(4, 1), U(2, 4), V(4, 0) MN TU 3 5 , NP UV 2 5 , PM VT 65 . So MNP TUV by SSS, and PMN VTU by CPCTC.4-6Review for MasteryTriangle Congruence: CPCTC continuedCopyright by Holt, Rinehart and Winston.14Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.53Holt GeometryAll rights reserved.LESSONA coordinate proof is a proof that uses coordinate geometry and algebra. In a coordinate proof, the first step is to position a figure in a plane. There are several ways you can do this to make your proof easier.Positioning a Figure in the Coordinate PlaneKeep the figure in0XY22Quadrant I by using the origin as a vertex.Center the figure 0XY223 3at the origin.Center a side of the 0XY33 3figure at the origin.Use one or both axes0XY33as sides of the figure.Position each figure in the coordinate plane and give the coordinates of each vertex.
XY
XY
1.a square with side lengths of 6 units2.a right triangle with leg lengths of 3 units and 4 units Possible answer on graph above. Possible answer on graph above.
XY
XY
3.a triangle with a base of 8 units and 4.a rectangle with a length of 6 units and a height of 2 unitsa width of 3 units Possible answer on graph above. Possible answer on graph above.4-7Review for MasteryIntroduction to Coordinate ProofCopyright by Holt, Rinehart and Winston.54Holt GeometryAll rights reserved.LESSONYou can prove that a statement about a figure is true without knowing the side lengths. To do this, assign variables as the coordinates of the vertices.
XYDCPosition each figure in the coordinate plane and give the coordinates of each vertex.
XYTS
XYK KKK5.a right triangle with leg lengths s and t 6.a square with side lengths k Possible answer on graph above. Possible answer on graph above.
XYWW
XYHB7.a rectangle with leg lengths and w 8.a triangle with base b and height h Possible answer on graph above. Possible answer on graph above.9.Describe how you could use the formulas for midpoint and slope to prove the following.Given: HJK, R is the midpoint of_HJ , S is the midpoint of_JK .Prove:_RS _HK Possible answer: Use the midpoint formula to find the coordinates of the midpoints R and S. Then use the coordinates and the formula for slope to find the slopes of_RS and_HK.4-7Review for MasteryIntroduction to Coordinate Proof continueda right triangle with leg lengths c and dCopyright by Holt, Rinehart and Winston.55Holt GeometryAll rights reserved.LESSONTheorem ExamplesIsosceles Triangle TheoremIf two sides of a triangle are congruent, then the angles opposite the sides are congruent.4 32If_RT _RS , then T S.Converse of Isosceles Triangle TheoremIf two angles of a triangle are congruent, then the sides opposite those angles are congruent.. -,If N M, then_LN _LM .You can use these theorems to find angle measures in isosceles triangles.Find mE in DEF.mD mEIsosc. Thm. $ %&X X 5x (3x + 14)Substitute the given values.2x 14Subtract 3x from both sides.x 7Divide both sides by 2.Thus mE 3(7) 14 35.Find each angle measure. ! #"
2 01
1.mC 512.mQ 47 '*(XX ,- .XX3.mH 724.mM 60Review for MasteryIsosceles and Equilateral Triangles 4-8Copyright by Holt, Rinehart and Winston.56Holt GeometryAll rights reserved.LESSON4-8Equilateral Triangle Corollary If a triangle is equilateral, then it is equiangular.(equilateral equiangular )Equiangular Triangle CorollaryIf a triangle is equiangular, then it is equilateral.(equiangular equilateral )If A B C, then_AB _BC _CA .You can use these theorems to find values in equilateral triangles.Find x in STV.STV is equiangular.Equilateral equiangular 34 6X(7x 4) 60The measure of each of an equiangular is 60.7x 56Subtract 4 from both sides.x 8Divide both sides by 7.Find each value. 12 3N $% &X5.n 126.x 33 34 6RR -. ,Y Y7.VT 188.MN 13Review for MasteryIsosceles and Equilateral Triangles continued!" #Copyright by Holt, Rinehart and Winston.15Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.57Holt GeometryAll rights reserved.LESSONReview for MasteryPerpendicular and Angle Bisectors 5-1Theorem ExamplePerpendicular Bisector TheoremIf a point is on the perpendicular bisector of a segment, then it is equidistant, or the same distance, from the endpoints of the segment. !& '
"Given: is the perpendicular bisector of_FG .Conclusion: AF AGThe Converse of the Perpendicular Bisector Theorem is also true. If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.You can write an equation for the perpendicular bisector of a segment. Consider the segment with endpoints Q (5, 6) and R (1, 2).Step 1 Find the midpoint of_QR .Step 2 Find the slope of the bisector of_QR . x1 x2 ______ 2,y1 y2 ______ 2 5 1 _______ 2 ,6 2 _____ 2
y2 y1 ______ x2 x1 2 6 ________ 1 (5) Slope of_QR (2, 4) 2 __ 3 So the slope of the bisector of_QR is3 __ 2 .Step 3 Use the point-slope form to write an equation.y y1 m (x x1)Point-slope formy 4 3 __ 2 (x 2)Slope 3 __ 2 ; line passes through (2, 4), the midpoint of_QR .Find each measure.2M4 31614 6 $T" !2.54# *+3X15X3( ,1. RT 162. AB 53. HJ 7Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints.4. A (6, 3), B (0, 5)5. W (2, 7), X (4, 3)y 1 3 __ 4 (x 3) y 5 3 __ 2 (x 1)Each point on is equidistant from points F and G.Copyright by Holt, Rinehart and Winston.58Holt GeometryAll rights reserved.LESSONReview for MasteryPerpendicular and Angle Bisectors continued 5-1Theorem ExampleAngle Bisector TheoremIf a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
-0,.Given: ___ MP is the angle bisector of LMN.Conclusion: LP NPConverse of the Angle Bisector TheoremIf a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. -0,.Given: LP NPConclusion: ___ MP is the angle bisector of LMN.Find each measure. ' &%(
2431
8:7 9X X6.EH 7.mQRS 8.mWXZ 23 52 35Use the figure for Exercises 911. ( *+,9.Given that __ JL bisects HJK and LK 11.4, find LH. 11.4 10.Given that LH 26, LK 26, and mHJK 122, find mLJK. 61 11.Given that LH LK, mHJL (3y 19), and mLJK (4y 5), find the value of y. 14Point P is equidistant from sides __ ML and ___ MN .LMP NMPCopyright by Holt, Rinehart and Winston.59Holt GeometryAll rights reserved.LESSON4. 13-20Theorem ExampleCircumcenter TheoremThe circumcenter of a triangle is equidistant from the vertices of the triangle.Given:_MR ,_MS , and_MT are4. 13-20the perpendicular bisectors of NPQ.Conclusion: MN MP MQIf a triangle on a coordinate plane has two sides that lie along the axes, you can easily find the circumcenter. Find the equations for the perpendicular bisectors of those two sides. The intersection of their graphs is the circumcenter. _HD,_JD, and_KD are the perpendicular bisectors of EFG. &+% '( *$
Find each length.1. DG2. EK 19 173. FJ4. DE 15 19Find the circumcenter of each triangle.5.XY#$ /
6.XY,(0, 5)-(8, 0) /(0, 0)34(2, 3) ( 4, 2.5)Review for MasteryBisectors of Triangles 5-2The point of intersection of_MR ,_MS , and_MT is called the circumcenter of NPQ.Perpendicular bisectors _MR ,_MS , and_MT are concurrent because they intersect at one point.Copyright by Holt, Rinehart and Winston.60Holt GeometryAll rights reserved.LESSON' *(!Theorem ExampleIncenter TheoremThe incenter of a triangle is equidistant from the sides of the triangle.Given:_AG ,_AH , and_AJ are'$#"*(!the angle bisectors of GHJ.Conclusion: AB AC AD _WM and_WP are angle bisectors of MNP, and WK 21. +- 0.7
Find mWPN and the distance from W to_MN and_NP .mNMP 2mNMWDef. of bisectormNMP 2(32) 64Substitute.mNMP mN mNPM 180 Sum Thm.64 72 mNPM 180Substitute.mNPM 44Subtract 136 from each side.mWPN 1 __ 2 mNPMDef. of bisectormWPN 1 __ 2 (44) 22Substitute.The distance from W to_MN and_NP is 21 by the Incenter Theorem. _PC and_PD are angle bisectors of CDE. Find each measure. #$10%
7.the distance from P to_CE8.mPDE 9 34 _KX and_KZ are angle bisectors of XYZ. Find each measure. : 8+9
9.the distance from K to_YZ10.mKZY 33 52.55-2Review for MasteryBisectors of Triangles continuedThe point of intersection of_AG ,_AH , and_AJ is called the incenter of GHJ.Angle bisectors of GHJ intersect at one point.Copyright by Holt, Rinehart and Winston.16Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.61Holt GeometryAll rights reserved.LESSON* ! #(.'"Theorem ExampleCentroid TheoremThe centroid of a triangle is located2 __ 3 of the distance from each vertex to the midpoint of the opposite side.* ! #(.'"Given:_AH ,_CG , and_BJ are medians of ABC.Conclusion: AN 2 __ 3 AH, CN 2 __ 3 CG, BN 2 __ 3 BJIn ABC above, suppose AH 18 and BN 10. You can use the Centroid Theorem to find AN and BJ.AN 2 __ 3 AHCentroid Thm.BN 2 __ 3 BJCentroid Thm.AN 2 __ 3 (18)Substitute 18 for AH.10 2 __ 3 BJSubstitute 10 for BN.AN 12Simplify.15 BJSimplify.In QRS, RX 48 and QW 30. Find each length. 81 3:9721. RW2. WX 32 163. QZ4. WZ 45 15In HJK, HD 21 and BK 18. Find each length. *+(#$"%5. HB6. BD 14 77. CK8. CB 27 95-3Review for MasteryMedians and Altitudes of TrianglesThe point of intersection of the medians is called the centroid of ABC. _AH ,_BJ , and_CG are medians of a triangle. They each join a vertex and the midpoint of the opposite side.Copyright by Holt, Rinehart and Winston.62Holt GeometryAll rights reserved.LESSON%* ,$"#+Find the orthocenter of ABC with vertices A (3, 3), B (3, 7), and C (3, 0).Step 1 Graph the triangle. XY"#!
Step 2 Find equations of the lines containing two altitudes.The altitude from A to_BC is the horizontal line y 3.The slope of__AC 0 3 ________3 (3) 1 __2 , so the slope of the altitudefrom B to_AC is 2. The altitude must pass through B(3, 7). y y1 m(x x1) Point-slope form y 7 2(x 3)Substitute 2 for m and the coordinates of B (3, 7) for (x1, y1).y 2x 1Simplify.Step 3Solving the system of equations y 3 and y 2x 1, you find that the coordinates of the orthocenter are (1, 3).Triangle FGH has coordinates F (3, 1), G (2, 6), and H (4, 1).9.Find an equation of the line containing theXY'( &
altitude from G to_FH .x 2 10.Find an equation of the line containing the altitude from H to_FG . y x 5 11.Solve the system of equations from Exercises 9 and 10 to find the coordinates of the orthocenter.(2, 3)Find the orthocenter of the triangle with the given vertices. 12.N (1, 0), P (1, 8), Q (5, 0)13.R (1, 4), S (5, 2), T (1, 6) (1, 1) (3, 2)Review for MasteryMedians and Altitudes of Triangles continued 5-3The point of intersection of the altitudes is called the orthocenter of JKL. _JD ,_KE , and_LC are altitudes of a triangle. They are perpendicular segments that join a vertex and the line containing the side opposite the vertex.Copyright by Holt, Rinehart and Winston.63Holt GeometryAll rights reserved.LESSONA midsegment of a triangle joins the midpoints of two sides of the triangle. Every triangle has three midsegments.3# %2$Use the figure for Exercises 14._AB is a midsegment of RST.1.What is the slope of midsegment_AB and the slopeXY3(2, 3)!(1, 0)4(6, 1)"(3, 2)2(0, 3)223 0of side_ST ?1; 12.What can you conclude about_AB and_ST ?Since the slopes are the same,_AB _ST .3.Find AB and ST.AB 22 , ST 4 2 4.Compare the lengths of_AB and_ST . AB 1 __ 2 ST or ST 2ABUse MNP for Exercises 57.5. _UV is a midsegment of MNP. Find theXY.(4, 5)50(2, 1)6-(4, 7)43 3 0coordinates of U and V. U (1, 3), V (3, 2)6.Show that_UV _MN . The slope of_UV 1 __ 4 and the slope of _MN 1 __ 4 . Since the slopes are the same,_UV _MN .7.Show that UV 1 __ 2 MN. UV 17and MN 217 . Since 171 __ 2 (2 17 ), UV 1 __ 2 MN.5-4Review for MasteryThe Triangle Midsegment Theorem _RS is a midsegment of CDE.R is the midpoint of_CD . S is the midpoint of_CE .Copyright by Holt, Rinehart and Winston.64Holt GeometryAll rights reserved.LESSONTheorem ExampleTriangle Midsegment TheoremA midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side.1,.0-Given:_PQ is a midsegment of LMN.Conclusion:_PQ _LN , PQ 1 __ 2 LNYou can use the Triangle Midsegment Theorem to"(!# +*
find various measures in ABC. HJ 1 __ 2 AC Midsegment Thm. HJ 1 __ 2 (12)Substitute 12 for AC. HJ 6Simplify. JK 1 __ 2 AB Midsegment Thm. _HJ ||_ACMidsegment Thm.4 1 __ 2 ABSubstitute 4 for JK.mBCA mBJHCorr. Thm.8 ABSimplify.mBCA 35Substitute 35 for mBJH.Find each measure. (* '68 7
8.VX 239.HJ 54 10.mVXJ 92 11.XJ 27Find each measure.3%4#2$
12.ST 72 13.DE 22 14.mDES 48 15.mRCD 485-4Review for MasteryThe Triangle Midsegment Theorem continuedCopyright by Holt, Rinehart and Winston.17Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.65Holt GeometryAll rights reserved.LESSONIn a direct proof, you begin with a true hypothesis and prove that a conclusion is true. In an indirect proof, you begin by assuming that the conclusion is false (that is, that the opposite of the conclusion is true). You then show that this assumption leads to a contradiction.Consider the statement Two acute angles do not form a linear pair.Writing an Indirect ProofSteps Example1. Identify the conjecture to be proven. Given: 1 and 2 are acute angles.Prove: 1 and 2 do not form a linear pair.2.Assume the opposite of the conclusion is true.Assume 1 and 2 form a linear pair.3. Use direct reasoning to show that the assumption leads to a contradiction.m1 m2 180 by def. of linear pair.Since m1 90 and m2 90,m1 m2 180.This is a contradiction.4. Conclude that the assumption is false and hence that the original conjecture must be true.The assumption that 1 and 2 form a linear pair is false. Therefore 1 and 2 do not form a linear pair.Use the following statement for Exercises 14. #"!An obtuse triangle cannot have a right angle.1.Identify the conjecture to be proven. Given: ABC is an obtuse , B is an obtuse angle; Prove: ABC does not have a right angle.2.Assume the opposite of the conclusion. Write this assumption. Assume ABC does have a right angle. Let A be a right angle.3.Use direct reasoning to arrive at a contradiction. Possible answer: If A is a right angle, then mB mC 90. But mB > 90, since B is obtuse. So this is a contradiction.4.What can you conclude? The assumption that ABC does have a right angle is false. Therefore ABC does not have a right angle.Review for MasteryIndirect Proof and Inequalities in One Triangle 5-5Copyright by Holt, Rinehart and Winston.66Holt GeometryAll rights reserved.LESSONTheorem ExampleIf two sides of a triangle are not congruent, then the larger angle is opposite the longer side.879If WY XY, then mX mW.Another similar theorem says that if two angles of a triangle are not congruent, then the longer side is opposite the larger angle.Write the correct answer. 634
( *+
5.Write the angles in order from smallest 6.Write the sides in order from shortestto largest.to longest.V, S, T _JK ,_KH ,_HJ Theorem ExampleTriangle Inequality TheoremThe sum of any two side lengths of a triangle is greater than the third side length. ACB a b cb c ac a bTell whether a triangle can have sides with the given lengths. Explain.7.3, 5, 88.11, 15, 21No; 3 5 8, which is notYes; the sum of each pair of greater than the length of thelengths is greater than the length third side. of the third side.5-5Review for MasteryIndirect Proof and Inequalities in One Triangle continued _WY is the longest side.X is the largest angle.Copyright by Holt, Rinehart and Winston.67Holt GeometryAll rights reserved.LESSONTheorem ExampleHinge TheoremIf two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the included angle that is larger has the longer third side across from it.+ - '(,*If K is larger than G, then side_LM is longer than side_HJ .The Converse of the Hinge Theorem is also true. In the example above, if side_LM is longer than side_HJ , then you can conclude that K is larger than G. You can use both of these theorems to compare various measures of triangles.Compare NR and PQ in the figure at right. 012
3PN QRPR PRmNPR mQRPSince two sides are congruent and NPR is smaller than QRP, the side across from it is shorter than the side across from QRP.So NR PQ by the Hinge Theorem.Compare the given measures.389764
('&-,+
1.TV and XY2.mG and mL TV XY mG mL $ !"#
('&%
3.AB and AD 4.mFHE and mHFGAB AD mFHE mHFGReview for MasteryInequalities in Two Triangles 5-6Copyright by Holt, Rinehart and Winston.68Holt GeometryAll rights reserved.LESSONYou can use the Hinge Theorem and its converse to find a range of values in triangles.Use MNP and QRS to find the range of values for x..1230-
X
Step 1 Compare the side lengths in the triangles.NM SRNP SQmN mSSince two sides of MNP are congruent to two sides of QRS and mN mS, then MP QR by the Hinge Theorem.MP QRHinge Thm.3x 6 24Substitute the given values.3x 30Add 6 to each side.x 10Divide each side by 3.Step 2 Check that the measures are possible for a triangle.Since_MP is in a triangle, its length must be greater than 0.MP 0Def. of 3x 6 0Substitute 3x 6 for MP.x 2Simplify.Step 3 Combine the inequalities.A range of values for x is 2 x 10.Find a range of values for x.5.
X
6.2623(3X9)542 x 7 3 x 217. 14 10(2X6)1088.
X
3 x 57 0.6 x 75-6Review for MasteryInequalities in Two Triangles continuedCopyright by Holt, Rinehart and Winston.18Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.69Holt GeometryAll rights reserved.LESSONThe Pythagorean Theorem states that the following relationship exists among the lengths of the legs, a and b, and the length of the hypotenuse, c, of any right triangle.ACBa 2 b 2 c 2Use the Pythagorean Theorem to find the value of x in each triangle.
X
XXa 2 b 2 c 2Pythagorean Theorema 2 b 2 c 2x 2 62 92Substitute.x 2 42 (x 2)2x 2 36 81Take the squares.x 2 16 x 2 4x 4x 2 45Simplify.4x 12x 45x 3x 3 5 Find the value of x. Give your answer in simplest radical form.1.
X2.
Xx 12 x 29 3.
X4.
XXx 39x 405-7Review for MasteryThe Pythagorean TheoremTake the positive square root and simplify.Copyright by Holt, Rinehart and Winston.70Holt GeometryAll rights reserved.LESSONA Pythagorean triple is a set of three nonzero whole numbers a, b, and c that satisfy the equation a 2 b 2 c 2.You can use the following theorem to classify triangles by their angles if you know their side lengths. Always use the length of the longest side for c.Pythagorean Inequalities Theorem!"#CABIf c 2 a 2 b 2, then ABC is obtuse.!"#CABIf c 2 a 2 + b 2, then ABC is acute.Consider the measures 2, 5, and 6. They can be the side lengths of a triangle since 2 5 6, 2 6 5, and 5 6 2. If you substitute the values into c 2 a 2 b 2, you get 36 29. Since c 2 a 2 b 2, a triangle with side lengths 2, 5, and 6 must be obtuse.Find the missing side length. Tell whether the side lengths form a Pythagorean triple. Explain.5.
6.
10; yes; the three side lengths are95 ; no;95is not a whole nonzero whole numbers that number.satisfy a2 b2 c 2.Tell whether the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.7.4, 7, 9 8.10, 13, 169.8, 8, 11 yes; obtuse yes; acute yes; acute 10.9, 12, 15 11.5, 14, 2012.4.5, 6, 10.2 yes; right no yes; obtuse5-7Review for MasteryThe Pythagorean Theorem continuedPythagorean TriplesNot Pythagorean Triples3, 4, 5,5, 12, 132, 3, 46, 9, 117 mC 90 mC 90Copyright by Holt, Rinehart and Winston.71Holt GeometryAll rights reserved.LESSONTheorem Example45-45-90 Triangle TheoremIn a 45-45-90 triangle, both legs are congruent and the length of the hypotenuse is2times the length of a leg.
qiqi In a 45-45-90 triangle, if a legXX
Xqilength is x, then the hypotenuse length is x 2 .Use the 45-45-90 Triangle Theorem to find the value of x in EFG.Every isosceles right triangle is a 45-45-90 triangle. TriangleX& '%X
EFG is a 45-45-90 triangle with a hypotenuse of length 10.10 x 2Hypotenuse is 2times the length of a leg. 10 ___ 2 x 2 ____ 2 Divide both sides by 2 .52 xRationalize the denominator.Find the value of x. Give your answers in simplest radical form.1. X
2. X
x 17 2x 22 2 3.XX
4. X
qix 4 2x 25Review for MasteryApplying Special Right Triangles 5-8Copyright by Holt, Rinehart and Winston.72Holt GeometryAll rights reserved.LESSON5-8Theorem Examples30-60-90 Triangle TheoremIn a 30-60-90 triangle, the length of the hypotenuse is 2 multiplied by the length of the shorter leg, and the longer leg is 3 multiplied by the length of the shorter leg.
qi qi
In a 30-60-90 triangle, if the shorter leg
XqiXXlength is x, then the hypotenuse length is 2x and the longer leg length is x.Use the 30-60-90 Triangle Theorem to find the values
XY*( +of x and y in HJK.12 x 3Longer leg shorter leg multiplied by 3 . 12 ___ 3 xDivide both sides by 3 . 4 3 xRationalize the denominator.y 2xHypotenuse 2 multiplied by shorter leg.y 2(4 3 )Substitute 4 3for x.y 8 3Simplify.Find the values of x and y. Give your answers in simplest radical form.5.
XY6.
XYx 9; y 93x 2 3 ; y 47. XYqi8.
XYx 12 3 ; y 36 x 11 3 ; y 22 3 Review for MasteryApplying Special Right Triangles continued001_041_Go08an_RFM_TE.indd 18 3/23/07 10:43:46 AMCopyright by Holt, Rinehart and Winston.19Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.73Holt GeometryAll rights reserved.LESSONReview for MasteryProperties and Attributes of Polygons 6-1The parts of a polygon are named on the quadrilateral below. You can name a polygon by the number of its sides.A regular polygon has all sides congruent and all angles congruent. A polygon is convex if all its diagonals lie in the interior of the polygon. A polygon is concave if all or part of at least one diagonal lies outside the polygon. Types of Polygonsregular, convex irregular, convex irregular, concaveTell whether each figure is a polygon. If it is a polygon, name it by the number of sides.1. 2. 3.polygon; pentagon polygon; heptagon not a polygonTell whether each polygon is regular or irregular. Then tell whether it is concave or convex.4. 5. 6.irregular; convex regular; convex irregular; concaveNumber of Sides Polygon3 triangle4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10 decagonn n-gondiagonalvertexsideCopyright by Holt, Rinehart and Winston.74Holt GeometryAll rights reserved.LESSONReview for MasteryProperties and Attributes of Polygons continued 6-1The Polygon Angle Sum Theorem states that the sum of the interior angle measures of a convex polygon with n sides is (n 2)180.Convex PolygonNumber of SidesSum of Interior Angle Measures: (n 2)180quadrilateral 4 (4 2)180 360hexagon 6 (6 2)180 720decagon 10 (10 2)180 1440If a polygon is a regular polygon, then you can divide the sum of the interior angle measures by the number of sides to find the measure of each interior angle.Regular PolygonNumber of SidesSum of Interior Angle MeasuresMeasure of Each Interior Anglequadrilateral 4 360 360 4 90hexagon 6 720 720 6 120decagon 10 1440 1440 10 144The Polygon External Angle Sum Theorem states that the sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360. 1521451526314536063The measure of each exterior angle of a regular polygon with n exterior angles is 360 n. So the measure of each exterior angle of a regular decagon is 360 10 36.Find the sum of the interior angle measures of each convex polygon.7.pentagon8.octagon9.nonagon 540 1080 1260Find the measure of each interior angle of each regular polygon. Round to the nearest tenth if necessary. 10.pentagon11.heptagon12.15-gon 108 128.6 156Find the measure of each exterior angle of each regular polygon. 13.quadrilateral14.octagon 90 45Copyright by Holt, Rinehart and Winston.75Holt GeometryAll rights reserved.LESSON6-2Review for MasteryProperties of ParallelogramsA parallelogram is a quadrilateral with two pairs of parallel sides. All parallelograms, such as FGHJ, have the following properties. ' (& *^&'(*Properties of Parallelograms _FG _HJ _GH _JF Opposite sides are congruent.F H G J Opposite angles are congruent.mF mG 180mG mH 180mH mJ 180mJ mF 180Consecutive angles are supplementary. _FP _HP _GP _JP The diagonals bisect each other.Find each measure.1.AB2.mD !$" #CMCM !"# $
10 cm 70Find each measure in LMNP.3.ML4.LP 12 m 10 m5.mLPM6.LN 62 18 m - ., 01623210 m12 m9 m7.mMLN8.QN 32 9 m' (& *' (& *' (& *' (0& *Copyright by Holt, Rinehart and Winston.76Holt GeometryAll rights reserved.LESSON6-2Review for MasteryProperties of Parallelograms continuedYou can use properties of parallelograms to find measures.WXYZ is a parallelogram. Find mX. 7 :8 9XXmW mX 180 If a quadrilateral is a , then cons. are supp.(7x 15) 4x 180Substitute the given values.11x 15 180Combine like terms.11x 165Subtract 15 from both sides.x 15Divide both sides by 11.mX (4x) [4(15)] 60If you know the coordinates of three vertices of a parallelogram, you can use slope to find the coordinates of the fourth vertex.Three vertices of RSTV are R(3, 1), S(1, 5), and T(3, 6). Find the coordinates of V.Since opposite sides must be parallel, the rise and the run from S to R must be the same as the rise and the run from T to V.From S to R, you go down 4 units and right 4 units. So, from T to V, go down 4 units and right 4 units. Vertex V is at V(7, 2).You can use the slope formula to verify that_ST _RV . XY32643344CDEF is a parallelogram. Find each measure.9.CD10.EF # &$ %3Z4W85W1(9Z12) 36 36 11.mF12.mE 48 132The coordinates of three vertices of a parallelogram are given. Find the coordinates of the fourth vertex. 13.ABCD with A(0, 6), B(5, 8), C(5, 5) D(0, 3) 14.KLMN with K(4, 7), L(3, 6), M(5, 3) N(2, 4)Copyright by Holt, Rinehart and Winston.20Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.77Holt GeometryAll rights reserved.LESSONReview for MasteryConditions for Parallelograms 6-3You can use the following conditions to determine whether a quadrilateral such as PQRS is a parallelogram. 0 31 2Conditions for Parallelograms _QR _SP _QR _SP If one pair of opposite sides is and , then PQRS is a parallelogram. _QR _SP _PQ _RS If both pairs of opposite sides are , then PQRS is a parallelogram.P RQ SIf both pairs of opposite angles are , then PQRS is a parallelogram. _PT _RT _QT _ST If the diagonals bisect each other, then PQRS is a parallelogram.A quadrilateral is also a parallelogram if one of the angles is supplementary to both of its consecutive angles.65 115 180, so A is supplementary to B and D.Therefore, ABCD is a parallelogram. # "! $
Show that each quadrilateral is a parallelogram for the given values. Explain.1.Given: x 9 and y 42.Given: w 3 and z 312 34 1XYYX $%#&4W2(3Z25)2ZW7 QR ST 12; RS TQ 16; DE FC 10; mE 118 both pairs of opp. sides are . and mF 62, so E and F are supp. and_DE _FC ; one pair of opposite sides are and .0 31 20 31 20 31 20 31 24Copyright by Holt, Rinehart and Winston.78Holt GeometryAll rights reserved.LESSONYou can show that a quadrilateral is a parallelogram by using any of the conditions listed below.Conditions for Parallelograms Both pairs of opposite sides are parallel (definition). One pair of opposite sides is parallel and congruent. Both pairs of opposite sides are congruent. Both pairs of opposite angles are congruent. The diagonals bisect each other. One angle is supplementary to both its consecutive angles.& '% (, +- *EFGH must be a parallelogram JKLM may not be a parallelogram because both pairs of opposite because none of the sets of conditions sides are congruent.for a parallelogram is met.Determine whether each quadrilateral must be a parallelogram. Justify your answer.3.4. Yes; one pair of opp. sides is Yes; the diagonals bisect each and . other.5.6. No; none of the sets of conditions Yes; both pairs of opp. are . for a parallelogram is met. Show that the quadrilateral with the given vertices is a parallelogram by using the given definition or theorem.7.J(2, 2), K(3, 3), L(1, 5), M(2, 0)8.N(5, 1), P(2, 7), Q(6, 9), R(9, 3)Both pairs of opposite sides are parallel.Both pairs of opposite sides arecongruent. slope of_JK slope of_LM 5; NP QR 3 5 ; slope of_KL slope of_MJ 1 __ 2 PQ RN 25 6-3Review for MasteryConditions for Parallelograms continuedCopyright by Holt, Rinehart and Winston.79Holt GeometryAll rights reserved.LESSON6-4Review for MasteryProperties of Special ParallelogramsA rectangle is a quadrilateral with four right angles. A rectangle has the following properties.Properties of Rectangles(' +'(*+ is a parallelogram.*If a quadrilateral is a rectangle, then it is a parallelogram.(' +'*(+*If a parallelogram is a rectangle, then its diagonals are congruent.Since a rectangle is a parallelogram, a rectangle also has all the properties of parallelograms.A rhombus is a quadrilateral with four congruent sides. A rhombus has the following properties.Properties of Rhombuses2 34 11234 is a parallelogram.If a quadrilateral i