Untitled

M N

23 Geometry

1·1 Understanding Points, Lines, and Planes

Draw and label each of the following.

1. a segment containing the points A and 8

2. a ray with endpoint M that passes through N

3. a plane containing a line segment with endpoints X and Y

4. three coplanar lines intersecting ih one point.

Name each of the following.

5. three coplanar points A,B,C

g $

A 8

• )------;

/y

*6. a line contained in neither plane

7. a segment contained in plane R FG

8. a line contained in both planes 8E

1·2 Measuring and Constructing Segments

Find the length of each segment.

A0011( I I c> I

-5

BC1 •1 • I I I I r-.-0 5

9. AB 10.8C 11.AC

3.5 1.5 5

12. Sketch, draw, and construct a segment congruent to EF.

E F•

E• F•

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y z

+

•

I CHAPTER 1 REVIEW CONTINUED I

13. B is between A and C. AC = 24 and BC = 11. Find AB. 13

26 14. Y is between X and Z.

Find XY. 14X 3X+ 2

M is the midpoint of AB. AM= 9x- 6, and BM = 6x + 27.

15. Find x. 16. Find AM. 17. Find BM.

11 93 93

1-3 Measuring and Constructing Angles

18. Name all the angles in the diagram.

LEFG, LEFH, LHFG o( •

E F G

Classify each angle by its measure.

19. m L XYZ = 89° 20. m L PQR = 150° 21.m L BRZ= goo

acute obtuse right

22. MT bisects L LMP, m L LMT, = (3x + 12)0 , and

m L TMP, = (6x- 24)0

Find m L LMP.

23. Use a protractor and a straightedge to draw an 80° angle. Then bisect the angle.

//

//

//

//

//

/

1·4 Pairs of Angles

Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent.

24. L2 and L3 only adjacent

adjacent and form a25. L3 and L4 linear pair

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'

,

I CHAPTER 1 REVIEW CONTINUED I

26. L3 and L1 not adjacent

If mLA = (7x- 12) 0

find the measure of each of the following.

27. supplement of L A 28. complement of L A

192- 7x 102- 7x

1·5 Using Formulas in Geometry

Find the perimeter and area of each figure.

29. 4x 30. i_j L

+12x

2X+ 83x-2

P = 9x + 6; A = 4x 2 + 16x P = 48x. A= 144x2

31. 32. 2x

x-3

P = 33 em; A= 50.06 cm 2 P = 6x - 6; A = 2x2 + 6x

33. Find the circumference and area of a circle with radius 9 in. Use then key on your calculator and round to the nearest tenth.

C 56.5 in.; A254.5 in 2

1-6 Midpoint and Distance in the Coordinate Plane

34. Find the coordinates of the midpoint of AB with endpoints A( -2, 6), andB(-4, -1).

(-3, 2.5)

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I CHAPTER 1 REVIEW CONTINUED I

35. Sis the midpoint of RT, R has coordinates (-4, -3) and S has coordinates(3, 5). Find the coordinates of T.

(10, 13)

36. Using the distance formula, find PO and RS to the nearest tenth. Then determine if PO :::.: RS.

\126 = 5.1; yes; PQ:::::: RS

37. Using the Distance Formula and the Pythagorean Theorem, find the distance, to the nearest tenth, from M(4, -3)to N(-5, 2).

v106:::::: 10.3

1-7 Transformations in the Coordinate Plane

Identify the transformation.Then use arrow notation to describe the

38.

FQD'

c·

39.K'DL'

transformation.

J'A BAo' 8K

E C M

L

D

reflection: ABCDE A'B'C'D'E'

rotation 90°; JKLM J'K'L'M'

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27 Geometry

I CHAPTER 1 REVIEW CONTINUED I

40. Find the coordinates for the image of figure JKLM after the translation (x, y) (x- 1, y + 2). Graph the image.

J'(-6, 6),K'( -3, 8), L'(O, 6) M'(-3, 0)

41. A figure has vertices at A(2, 4), 8( -5, 1) and C(O, -3). After a transformation, the image of the figure has vertices at A'(5, 6), 8'( -2, 3), and C'(3, -1). Graph the preimage and image. Then, identify the transformation.

Transformation: (x, y)(X+ 3, y+ 2)

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Student Morning Afternoon

Ann 2.4 1.9

Betsy 3.1 2.7

Carla 4.0 3.9

Denise 2.7 2.8

Ellen 2.2 2.0

. . - . -

H Chapter Review

2·1 Using Inductive Reasoning to Make Conjectures

Find the next term in each pattern.

1. 6, 12, 18, ... 2. January, April, July, ...

24 October

3. The table shows the score on a reaction time test given to five students in both the morning and afternoon. The lower scores indicate a faster reaction time. Use the table to make a conjecture about reaction times.

The scores for the afternoon test were lower, indicating a faster reaction time as compared to the morning test.

4. Show that the conjecture "If a number is a multiple of 5, then it is an odd number'' is false by finding a counterexample.

(10, 20, 30) are counterexamples

2·2 Conditional Statements

5. Identify the hypothesis and conclusion of the conditional statement "Two angles whose sum is goo are complementary angles".

Hypothesis:Two angles whose sum is goo.Conclusion:The angles are complementary.

Write a conditional statement from each of the following.

If a number is an even number, then it is an integer.

If an angle measurers90°, then the angle is a

7. An angle that measures goo is a right angle. right angle.

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44 Geometry

I CHAPTER 2 REVIEW CONTINUED

Determine if each conditional is true. If false, give a counterexample.

8. If an angle has a measure of goo, then it is an acute angle.

False, it is a right angle.

9. If 6x- 2 = 4x + 12, then x = 3. False, x = 7

10. Write the converse, inverse, and contrapositive of the statement "If a number is divisible by 4, then it is an even number." Find the truth value of each.

converse: If a number is an even number, then it is divisible by 4truth value: Finverse: If a number is not divisible by 4, tllen it is not an even number.truth value: Fcontrapositive: If a number is not an even number, then it is not divisible by 4.truth value: T

2·3 Using Deductive Reasoning to Verify Conjectures

11. Determine if the following conjecture is valid by the Law of Detachment.Given: Nicholas can watch 30 minutes of television if he cleans his room first. Nicholas cleans his room.Conjecture: Nicholas watches 30 minutes of television. Valid

12. Determine if the following conjecture is valid by the Law of Syllogism.Given: If a point A is on MN, then it divides MN into MA and AN. IfMA AN then A is the midpoint of MN.Conjecture: If a point is on MN, then A is the midpoint of MN.

No, it is not valid.

2·4 Biconditional Statements and Definitions

13. For the conditional "If two angles are complementary, then the sum of the measures is goo," write the converse and a biconditional statement.

Converse: If the sum of the measures of two angles is goo, then the two angles are conplementary.Biconditional statement: Two angles are complementary if and only if the sum of their measures is goo.

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45 Geometry

I CHAPTER 2 REVIEW CONTINUED I

14. Determine if the biconditional "A point divides a segment into two congruent segments if and only if the point is the midpoint of the segment," is true. If false, give a counterexample.

True

2-5 Algebraic Proof

Solve each equation. Write a justification for each step.

15.m+3=-2 16. 3m- 4 = 20

-3 -3 Sub. Prop.of Equalitym = -5 Simplify.

+4 +4, Add.Prop. of Equality3m= 24,Simplify.

3;' = 234 Div.

Prop. of Equality m = 8 Simplify.

-2(_x2

) = -5(-

2), Mult Prop. of Equalityx = 10 Simplify.

Identify the property that justifies each statement.

18. mL1mL2, so mL1 +mL3 = mL2 = mL3

19. MN PO, so POMN

Addition Property ofEquality

Symmetric Property ofCongruence

20. AB = CD and CD = EF,so AB= EF

21. mLA = mLA

Transitive Property ofEquality

Reflexive Property ofEquality

2-6 Geometric Proof

22. Fill in the blanks to complete the two-column proof.. mLMOP = mLROP = goo

Grven: L1 L4

Prove: L2L3

Proof:

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Statements Reasons

LMOP ::::: LNOQ Given

, mLMOP = mLNOQ Definition of Congruent Angles

mLMON + mLNOP = mLMOPmLPOQ + mLNOP = mLNOQ

Angle Addition Postulate

mLMON + mLNOP = mLPOQ +mLNOP

Transitive Property of Equality

mLMON = mLPOQ Subtraction Property of Equality

LMON::::: LPOQ Definition of Congruent Angles

I CHAPTER 2 REVIEW CONTINUED I

---- ·- -··-i Statements ReasonsI'--- ,-- -

1. mLMOP = m LROP =goo 1. GivenL1 :.= L4

J 2. mL 1 = m L4

1 m-L1 mL2

2. Definition of Congruent Angles-

= mLMOP 3. Angle Addition Postulatej

3· mL3 + mL4 = mLMOP----- - ....

1 4. mL1 + mL2 = mL3 + mL4 4. Transitive Property of Equality

m_1 + m L2 = m L 3 + m L1 5. SubstitutionI

L2 _mL_3

j 6. Subtraction..Proprty of Equality

23. Use the given plan to write a two-column proof.

Given: LMOPL NOQ

Prove: L MON = LPOQ

Plan: By the definition of angle 0 congruence, mLMOP = mLNOQ.Use the angle addition postulate to show that mLMOP = mLMON +mLNOP. Show a similar statement for LNOQ. Use the given fact to equate mLMON + mLNOP and mLPOQ + mLNOP. The subtraction property of equality allows you to show m L MON = mLPOQ. Use the definition ofcongruent triangles to establish what needs to be proved.

--

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66 Geometry

3-1 Lines and Angles

Identify each of the following.

Sample answer: BC

1. a pair of parallel segments and AD

Sample answer: AB

2. a pair of perpendicular segments and BC

3. a pair of skew segments 4. a pair of parallel planes

Sample answer: AE and CD Sample answer: planeBHGC and plane AEFD

Give an example of each angle pair.

5. alternate interior angles

L2 and L7, L6 and L3

L1 and L3, L2 and L4,6. corresponding angles L5 and L7, L6 and LB

7. alternate exterior angles 8. same-side interior angles

L1 and LB, L4 and L5 L2 and L3, L6 and L7

3-2 Angles Formed by Parallel Lines and Transversals

Find each angle measure.

9. 10. 11.

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67 Geometry

c

I CHAPTER 3 REVIEW CONTINUED I

3·3 Proving Lines Parallel

Use the given information and the theorems and postulates you have learned to show that a II b.

Converse c.:rf Alternate Interior

12. mL2 = mL7 .l\ng!es Theorem

Same-side interior angles

13. m L 3 x mL7 = 180° have a sum of 180 degrees.

14. mL4 = (4x + 34t,mL7 = (7x- 38)0

, x = 2415. mL1mL5

Corresponding anglesare congruent.

Corresponding anglesare congruent.

16. If L1 :::: L 2, write a paragraph proof to show that DC II AB.

It is given that L1 :.:::: L2, and since vertical angles are congruent, L2 :.:::: L3. By the transitive property, L1 :::: L3 and thereforeDC II AB because when two lines are cut

A

.- 1r--- ----•tby a transversal, and correspondingangles are congruent, the lines are parallel

8

(corresponding angles congruent postulate).

3-4 Perpendicular Lines r17. Complete the two-column proof below.

Given: r _l_ v, L 1 :::: L2

Prove: r _l_ s

w

.-- ---- - ----Statements Reasons

lo---

1 . r _l_ v, L1L2

I 2. s II vI 1. Given

2. Converse of correspondingangles are congruent J

Ij 3. r _l_ s

1

Perpen icular transversal theore

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68

l CHAPTER 3 REVIEW CONTINUED I

3-5 Slopes of Lines

Use the slope formula to determine the slope of each line.

1a. cr 19. A83

4 4

20. Ef: 21. Dai 22 3

Find the slope of the line through the given points.

22. R(2, 3) and S(4, 9) 33

23. C(4, 6) and 0(8, 3) 4

24. H( -8, 7) and /(2, 7) zero 25. S(4, 0) and T(3, 4) -4

Graph each pair of lines and use their slopes to determine if they are parallel, perpendicular, or neither.

26. Co and AB for A(3, 6), 8(6, 12), C(4, 2), and 0(5, 4)

27. rM and NP for L( -6, 1), M(1, 8), N(-1, -2), and P(-3, 0)

parallel

Copyright © by Hoa, Rinahart arod ·'VinstonAll rights reserved.

X

perpendicular

Geometry

5

I CHAPTER 3 REVIEW CONTINUED I

28. Ps and Rs for P(6, 6), 0(5, 7), R(5, -2), and S(7, 2)

29. GH and FJ for F( -5, -4), G( -3, -10), H(-5, 0), and J( -8, -1)

neither

X

neither

3·6 Lines in the Coordinate Plane

Write the equation of each line in the given form.

30. the line through (1, -1) and ( -3, -3) in slope-intercept form y=x-2

231. the line through ( -5, -6) with slopein point-slope form Y + 6

= 5(x + S)

32. the line with y-intercept 3 through the point (4, 1) in slope-intercept form

33. the line with x-intercept 5 and y-intercept -2in slope-intercept form

y= 2 x- 2

Graph each line.

34. y= -3x+ 2

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35.x= 4

) y

69

36. y + 2 = -1(x - 3)

·:····=··)..)....

Geometry

I CHAPTER 3 REVIEW CONTINUED I

Write the equation of each line.

37.

X= -3

38.

y=3

39.

y = -:-2x + 1

Determine whether the lines are parallel, intersect, or coincide.

4x+5y=10

40. Y = - x+ 2

coincide

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_y=-7x+141 y= -7x- 3

parallel

70

42 y= 6x- 5·4x + 6y= 8

intersect

Geometry

.

93 Geometry

L

a Chapter Review

4·1 Classifying Triangles w

Classify each triangle by its angle measure.

1. 6XYZ 2.6XYW 3.6XZW

Acute Equiangular Obtuse X y

Classify each triangle by its side lengths.

4. 60EF 5. 6DEG 6.6EFG &Equilateral Scalene Isosceles F E

4·2 Angle Relationships in Triangles

Find each angle measure.

7. mLACB B.mLK K

9. A carpenter built a triangular support structure for a roof. Two of the angles of the structure measure 32.5° and 47.5°. Find the measureof the third angle.

4·3 Congruent Triangles

M A

N

Given l:lABC :::: l:lXYZ. Identify the congruent corresponding parts.

10. BC:::: YZ

12. LA- LX

11. zx-13. L Y:::: LB

Given f:lJKL - f:lPQR. Find each value.

14.x 15. RP

14 15

L K R Q

2y-1

J p

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94 Geometry

I CHAPTER 4 REVIEW CONTINUED I

16. Given: e II k; 80 CD; A8 :::: AC; AD l_ CB; AD l_ XW; L XAC WAB

Prove: 6A80 6ACDe x w

k y

c

Statements Reasons

1. 80 CO ; AB :::: AC; 1. Given

2. ADAD 2. Reflexive Property ofCongruence

3. e 11 k; AD 1_ C8; AD 1_ xw 3. Given

4. LADB and LADC are right angles.

4. Def. of l_ lines

5. L ADB LADC 5. Rt. L :::: Thm

6. LXAC ==: LWAB 6. Given

7. L XAC :::::: L ACO; L WAB LABO

7. Parallel lines cut by a transversal, alternate interior angles are congruent.

8. LACD :::: LABD 8. Transitive Property ofCongruence

9. LCAO:::: LBAD 9. Third Angles Theorem

10. b.ABD :::: b.ACD 10. Def of Congruent Triangles

4·4 Triangle Congruence: SSS and SAS

17. Given that HIJK is a rhombus, use SSS to explainwhy 6HIL 6JKL.

HI:::: JK by the definition of a rhombus.HL :::: JL and Ll:::: LK because diagonals of a rhombus bisect each other.Therefore, b.HIL ::::b.JKL by SSS.

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Statements Reasons

1. NR PR; MR :=:-OR 1. Given

2. LMRN -· LQRP 2. Vertical angles are congruent

3. L.MNR::::: L.QPR 13. SAS

Statements Reasons

1. SD bisects LABC andLADC

1. Given

2. LASD :::= LCBD, LADS :::= LCDS

2. Definition of angle bisector

3. SD= SD 3. Reflexive propertyof congruent angles

4. L..ASD :::= b..CSD 4.ASA

95 Geometry

I CHAPTER 4 REVIEW CONTINUED I

18. Given: NR ::::: PR; MR QR MN

Prove: L.MNR L.QPR

pQ

4-5 Triangle Congruence: ASA, AAS, and HL

Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know.

19. L.HIK L.JIK H 20. L.PQR L.RSP

No; HI :::= Jl Yes

J s

21. Use ASA to prove the triangles congruent. A c

Given: 80 bisects LABC and LADC

Prove: L.ABD ::::: 6CBD

D

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i ; :

96 Geometry

I CHAPTER 4 REVIEW CONTINUED I

4·6 Triangle Congruence: CPCTCX

22. Given: UZYZ, VZ == XZ

Prove: XY VU v

Statements Reasons

1. UZ== YZ, VZ== XZ 1. Given

2. LUZV== LYZX 2. Vertical angles are congruent

3. !;:,.UZV == /;:,.YZX 3. SAS

4. XY:::: VU 4. CPCTC

4·7 Introduction to Coordinate Proof Answers will vary. SamplePosition each figure in the coordinate plane. answers shown.

23. a right triangle with legs 3 and4 units in length

24. a rectangle with sides 6 and8 units in length

. .-4 -···· :. ;_ _ _ ..:._

(0 3)....- .....;. . !...........

...

-8

..6.._------'-----f;J-····- - -···----- ······-·-· ....

j _,: ..:.....:...............: .. .:tb :r!}Fi-(s;: Q) X

-1...... 1-H-d+H"2 "4·-= "ij"' h"

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JJ<o- a ) c ) Jta tc

97 Geometry

I CHAPTER 4 REVIEW CONTINUED I

4·8 Isosceles and Equilateral Triangles

25. Assign coordinates to each vertex and write a coordinate proof.

Given: rectangle ABCD with diagonals intersecting at z Y

Prove: CZ = DZ

r ) ------ (0, c)The coordinates of z are \ a , because

diagonals of rectangles bisect each other, meaning they intersect at each other's midpoints. By the midpoint formula the coordinates of z are

{Q.+a O+c)\ 2 ' 2 .

CZ = (O--- -a-)2__+_(c -- ----)2- = J±a 2 + tc2

...------- Therefore,DZ =

2 + (0-2 = 2 + 2

CZ = DZ, which means CZ:::: DZ.

-- a

Find each angle measure.

26. mL B A 27. mLHEF E

c 8

28. Given: b.PQR has coordinates P(O, 0), Q(2a, 0), and R(a, aV3)

Prove: 6PQR is equilateral. y

PQ = v'(o - 2a)2 + (0 - 0)2 = 2a

QR = V(2a - a)2+ (0 - av3)2 =

2a:.. : ·'··{2a;--o)· xI I I I I

RP = Y(a - 0)2 + (aV3 - 0)2 = 2a

Since PQ = QR = RP, t6.PQR is equilateral.

8 ' 12 61 :Q

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117 Geometry

H I Chapter Review

5-1 Perpendicular and Angle Bisectors

Find each measure.

1. BC 2.RS 3.XW

B5.6

0

c w z y

11.2 43 27

4. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints K(10, 3) and L( -2, -5).

5·2 Bisectors of Triangles

5. NP, OP, MP are the perpendicular bisectors of i'::.JKL. Find PK and JM.

KPK= 9.9JM = 7.3

6. DH and OG are angle bisectors of i'::.EGH. Find mLEHD and the distance from 0 to GH.

G mLEHD = 44oDH= 42

E

7. Find the circumcenter of i'::.JKL with vertices J(O, 6), K(B, 0) and L(O, 0).

(4, 3)

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. G ;:

118 Geometry

I CHAPTER 5 REVIEW CONTINUED I

5·3 Medians and Altitudes of Triangles

8. Nathan cuts a triangle with vertices at coordinates (0, 4), (6, 0), and(3, -2) from a piece of grid paper. At what coordinates should he place the tip of a pen to balance the triangle? ( 2 )3,3

9. Find the orthocenter of 6WYX with vertices W(1, 2), X(7, 2), and Y(3, 5).

5·4 The Triangle Midsegment Theorem

10. Find BA, JC, and mLFBAin 6KCJ.

11. What is the distance MP across the lake?

NBA = 29JC = 134mLFBA = 40° 102 meters

K

12. Write an indirect proof that an equiangular triangle can not have an obtuse angle.

may vary.I •An equiangular triangle1 Prove: The triangle does not have an obtuse angleI 2. Assume the equiangular triangle has an obtuse angle

3. An equilateral triangle, by definition, is a triangle whose angles are all congruent. Since the sum of the angles in a triangle measure180°, then each angle in an equiangular triangle must measure 60°. jAn obtuse angle, by definition, is an angle whose measure is greater!than goo. Since 60° is less then 90°, none of the angles are obtuse. 1

I

4. Original conjecture is true. An equiangular triangle cannot have an j

obtus e an le.

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I CHAPTER 5 REVIEW CONTINUED I

5·5 Indirect Proof and Inequalities in One Triangle

13. Write the angles of 6QRSin order from smallest to largest.

R

41

Q 62.4 s

14. Write the sides of 6ABC in order from shortest to longest.

c

8 A

mLRSQmLRQSmLORS AC, AB, BC

Tell whether a triangle can have sides with given lengths. Explain.

15. 7.7, 9.4, 16.1 16. 3r, r + 4, r 2 when r = 5

Yes, the sum of each pair of lengths is greater than the third length.

No, the sum of the pair of lengths is not greater than the third length in all cases.

17. The distance from Tyler's house to the library is 3 miles. The distance from his home to the park is 12 miles. If the three locations form a triangle, whatis the range of distance from the library to the park?

5·6 Inequalities in Two Triangles

9 < d< 15 miles

18. Compare KL andNP.

N

19. Compare mLBAD andmLCAD.

A

20. Find the range of values of x.

Bc

v

NP>KL mLBAD > mLCAD 1.5 <X< 19

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