Organizer N - mryildiz.weebly.commryildiz.weebly.com/uploads/1/7/2/0/17209270/chapter_1-5_answers.pdf · J is the midpoint of ... Items 9–11. Write the indicated type of statement

64 Chapter 1 Foundations for Geometry

1. Draw and label plane N containing two lines that intersect at B.

Use the figure to name each of the following.

2. four noncoplanar points 3. line containing B and E

4. The coordinate of A is -3, and the coordinate of B is 0.5. Find AB.

5. E, F, and G represent mile markers along a straight highway. Find EF.

6. J is the midpoint of −

HK. Find HJ, JK, and HK.

Classify each angle by its measure.

7. m∠LMP = 70° 8. m∠QMN = 90° 9. m∠PMN = 125°

10. �� TV bisects ∠RTS. If the m∠RTV = (16x - 6)° and m∠VTS = (13x + 9)°, what is the m∠RTV?

11. An angle’s measure is 5 degrees less than 3 times the measure of its supplement. Find the measure of the angle and its supplement.

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

12. ∠2 and ∠3 13. ∠4 and ∠5 14. ∠1 and ∠4

15. Find the perimeter and area of a rectangle with b = 8 ft and h = 4 ft.

Find the circumference and area of each circle to the nearest tenth.

16. r = 15 m 17. d = 25 ft 18. d = 2.8 cm

19. Find the midpoint of the segment with endpoints (-4, 6) and (3, 2) .

20. M is the midpoint of −

LN. M has coordinates (-5, 1) , and L has coordinates (2, 4) . Find the coordinates of N.

21. Given A(-5, 1) , B(-1, 3) , C(1, 4) , and D(4, 1) , is −

AB � −

CD? Explain.

Identify each transformation. Then use arrow notation to describe the transformation.

22. 23.

24. A designer used the translation (x, y) → (x + 3, y - 3) to transform atriangular-shaped pin ABC. Find the coordinates and draw the image of �ABC.

2. Possible answer: D, E, C, A3. Possible Answer: �� BE

3.5

14

9; 9; 18

acute rt. obtuse

74°

133.75°; 46.25°

only adj. adj. and a lin. pair

not adj.

P = 24 ft; A = 32 ft 2

16. 94.2 m; 706.9 m 2

17. 78.5 ft; 490.9 ft 2

8.8 cm; 6.2 cm 2

(-0.5, 4)

(-12, -2)no; AB ≈ 4.5; CD ≈ 4.2

180° rotation; QRS → Q′R′S′

reflection;WXYZ → W′X′Y′Z′

A′ (-2, -2) ; B′ (1, 1) ; C′ (2, -2)

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64 Chapter 1

C H A P T E R

1

OrganizerObjective: Assess students’ mastery of concepts and skills in Chapter 1.

Online Edition

ResourcesAssessment Resources

Chapter 1 Tests

• Free Response (Levels A, B, C)

• Multiple Choice (Levels A, B, C)

• Performance Assessment

Test & Practice Generator

KEYWORD: MG7 Resources

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134 Chapter 2 Geometric Reasoning

Find the next item in each pattern.

1. 2. 405, 135, 45, 15, …

3. Complete the conjecture “The sum of two even numbers is −−− ? . ”

4. Show that the conjecture “All complementary angles are adjacent” is false by finding a counterexample.

5. Identify the hypothesis and conclusion of the conditional statement “The show is cancelled if it rains.”

6. Write a conditional statement from the sentence “Parallel lines do not intersect.”

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

7. If two lines intersect, then they form four right angles.

8. If a number is divisible by 10, then it is divisible by 5.

Use the conditional “If you live in the United States, then you live in Kentucky” for Items 9–11. Write the indicated type of statement and determine its truth value.

9. converse 10. inverse 11. contrapositive

12. Determine if the following conjecture is valid by the Law of Detachment.Given: If it is colder than 50°F, Tom wears a sweater. It is 46°F today.Conjecture: Tom is wearing a sweater.

13. Use the Law of Syllogism to draw a conclusion from the given information.Given: If a figure is a square, then it is a quadrilateral. If a figure is a

quadrilateral, then it is a polygon. Figure ABCD is a square.

14. Write the conditional statement and converse within the biconditional “Chad will work on Saturday if and only if he gets paid overtime.”

15. Determine if the biconditional “B is the midpoint of −−

AC iff AB = BC” is true. If false, give a counterexample.

Solve each equation. Write a justification for each step.

16. 8 - 5s = 1 17. 0.4t + 3 = 1.6 18. 38 = -3w + 2

Identify the property that justifies each statement.

19. If 2x = y and y = 7, then 2x = 7. 20. m∠DEF = m∠DEF

21. ∠X � ∠P, and ∠P � ∠D. So ∠X � ∠D. 22. If −−

ST � −−

XY , then −−

XY � −−

ST.

Use the given plan to write a proof in each format.

Given: ∠AFB � ∠EFDProve: �� FB bisects ∠AFC.Plan: Since vertical angles are congruent, ∠EFD � ∠BFC. Use the Transitive Property to conclude that ∠AFB � ∠BFC. Thus �� FB bisects ∠AFC by the definition of angle bisector.

23. two-column proof 24. paragraph proof 25. flowchart proof

5

even

If 2 lines are ‖, then they do not intersect.

T

valid

Figure ABCD is a polygon.

F; B is not between A and C.

Trans. Prop. of =

Reflex. Prop. of =

Sym. Prop. of �Trans. Prop. of �

F

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134 Chapter 2


Online Edition


Chapter 2 Tests





Answers 4. Possible answer: ∠1 and ∠2 are comp.,

but not adj.

7. Possible answer:

9. If you live in Kentucky, then you live in the United States; T.

10. If you do not live in the United States, then you do not live in Kentucky; T.

11. If you do not live in Kentucky, then you do not live in the United States; F.

14. Conditional: If Chad works on Saturday, then he gets paid overtime. Converse: If Chad gets paid overtime, then he will work on Saturday.

16. 8 - 5s = 1 (Given); -5s = -7 (Subtr. Prop. of =); s = 1.4 (Div. Prop. of =)

17. 0.4t + 3 = 1.6 (Given); 0.4t = -1.4 (Subtr. Prop. of =); t = -3.5 (Div. Prop. of =)

18. 38 = -3w + 2 (Given); 36 = -3w (Subtr. Prop. of =); -12 = w (Div. Prop. of =)

23–25. See p. A13.KEYWORD: MG7 Resources

C H A P T E R

2


206 Chapter 3 Parallel and Perpendicular Lines

Identify each of the following.

1. a pair of parallel planes

2. a pair of parallel segments

3. a pair of skew segments

Find each angle measure.

4. 5. 6.

Use the given information and the theorems and postulates you have learned to show f ‖ g.

7. m∠4 = (16x + 20)°, m∠5 = (12x + 32)°, x = 3

8. m∠3 = (18x + 6)°, m∠5 = (21x + 18)°, x = 4

Write a two-column proof.

9. Given: ∠1 � ∠2, n ⊥ �

Prove: n ⊥ m

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

10. 11. 12.

13. Greg is on a 32-mile bicycle trail from Elroy, Wisconsin, to Sparta, Wisconsin. He leaves Elroy at 9:30 A.M. and arrives in Sparta at 2:00 P.M. Graph the line that represents Greg’s distance from Elroy at a given time. Find and interpret the slope of the line.

14. Graph � �� QR and � �� ST for Q(3, 3) , R(6, -5) , S(-4, 6) , and T(-1, -2) . Use slopes to determine whether the lines are parallel, perpendicular, or neither.

15. Write the equation of the line through (-2, -5) with slope - 3_4

in point-slope form.

16. Determine whether the lines 6x + y = 3 and 2x + 3y = 1 are parallel, intersect, or coincide.

−− AC ‖

−− DF

m = 7_2

m = 0

y + 5 = - 3_4

(x + 2)

intersect

m = 32_4.5

≈ 7.1;

Greg’s average speed was about 7.1 mi/h.

m = 4_5

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206 Chapter 3

C H A P T E R

3


Online Edition


Chapter 3 Tests





Answers 1. plane ABC ‖ plane DEF

3. Possible answer: −−

AB and −−

CF are skew.

4. Both labeled angles measure 57°.



7. m∠4 = 68°, and m∠5 = 68°, so ∠4 � ∠5. f ‖ g by the Conv. of Alt. Int. � Thm.

8. m∠3 = 78°, and m∠5 = 102°, so m∠3 + m∠5 = 180°. f ‖ g by the Conv. of Same-Side Int. � Thm.

9. 1. ∠1 � ∠2, n ⊥ (Given) 2. ‖ m (Conv. of Corr. � Post.) 3. n ⊥ m (⊥ Transv. Thm.)

14.

‖



288 Chapter 4 Triangle Congruence

1. Classify �ACD by its angle measures.

Classify each triangle by its side lengths.

2. �ACD 3. �ABC 4. �ABD

5. While surveying the triangular plot of land shown, a surveyor finds that m∠S = 43°. The measure of ∠RTP is twice that of ∠RTS. What is m∠R?

Given: �XYZ � �JKLIdentify the congruent corresponding parts.

6. −−

JL � −−−− ? 7. ∠Y � −−−− ? 8. ∠L � −−−− ? 9. −−

YZ � −−−− ?

10. Given: T is the midpoint of −−

PR and −−

SQ.Prove: �PTS � �RTQ

11. The figure represents a walkway with triangular supports. Given that

−− GJ bisects

∠HGK and ∠H � ∠K, use AAS to prove �HGJ � �KGJ

12. Given: −−

AB � −−

DC, 13. Given: −−

PQ ‖ −−

SR,

−− AB ⊥

−− AC , ∠S � ∠Q

−−

DC ⊥ −−

DB Prove: −−

PS ‖ −−

QRProve: �ABC � �DCB

14. Position a right triangle with legs 3 m and 4 m long in the coordinate plane. Give the coordinates of each vertex.

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

Given: Square ABCDProve:

−− AC �

−− BD

Find each value.

16. y 17. m∠S

18. Given: Isosceles �ABC has coordinates A(2a, 0) , B(0, 2b), and C(-2a, 0) . D is the midpoint of

−− AC , and E is the midpoint of

−− AB.

Prove: �AED is isosceles.

rt.

scalene isosc. scalene

77°

−− XZ ∠K ∠Z

−− KL

-5 44°

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288 Chapter 4

C H A P T E R

4


Online Edition


Chapter 4 Tests





Answers 10. 1. T is the mdpt. of

−− PR and

−− SQ .

(Given) 2.

−− PT �

−− RT ,

−− ST �

−− QT (Def. of

mdpt.) 3. ∠PTS � ∠RTQ (Vert. Thm.) 4. �PTS � �RTQ (SAS Steps

2, 3)

Answers 11. 1. ∠H � ∠K (Given) 2.

−− GJ bisects ∠HGK. (Given)

3. ∠ HGJ � ∠ KGJ (Def. of bisect) 4.

−− JG �

−− JG (Reflex. Prop. of �)

5. �HGJ � � KGJ (AAS Steps 1, 3, 4)

12. 1. −−

AB ⊥ −−

AC , −−

DC ⊥ −−

DB (Given) 2. ∠BAC and ∠CDB are rt. . (Def. of

⊥) 3. � ABC and � DCB are rt. �. (Def. of

rt. �) 4.

−− AB �

−− DC (Given)

5. −−

BC � −−

CB (Reflex. Prop. of �) 6. � ABC � �DCB (HL Steps 5, 4)

13. 1. −−

PQ ‖ −−

SR (Given) 2. ∠QPR � ∠SRP (Alt. Int. Thm.) 3. ∠S � ∠Q (Given) 4.

−− PR �

−− RP (Reflex. Prop. of �)

5. �QPR � �SRP (AAS Steps 2, 3, 4) 6. ∠SPR � ∠QRP (CPCTC) 7.

−− PS ‖

−− QR (Conv. of Alt. Int. Thm.)

14.

15, 18. See p. A17.


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370 Chapter 5 Properties and Attributes of Triangles

Find each measure.

1. KL 2. m∠WXY 3. BC

4. −−−

MQ, −−−

NQ, and −−

PQ are the 5. −−

EG and −−

FG are angle 6. In �XYZ, XC = 261, perpendicular bisectors of bisectors of �DEF. and ZW = 118. �RST. Find RS and RQ. Find m∠GEF and the Find XW, BW, and BZ. distance from G to

−− DF.

7. Find the orthocenter of �JKL with vertices J(-5, 2) , K(-5, 10) , and L(1, 4) .

8. In �GHJ at right, find PR, GJ, and m∠GRP.

9. Write an indirect proof that two obtuse angles cannot form a linear pair.

10. Write the angles of 11. Write the sides of �BEH in order from �RTY in order from smallest to largest. shortest to longest.

12. The distance from Arville to Branton is 114 miles. The distance from Branton to Camford is 247 miles. If the three towns form a triangle, what is the range of distances from Arville to Camford?

13. Compare m∠SPV 14. Find the range of and m∠ZPV. values for x.

15. Find the missing side length in the triangle. Tell if the side lengths form a Pythagorean triple. Explain.

16. Tell if the measures 18, 20, and 27 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

17. An IMAX screen is 62 feet tall and 82 feet wide. What is the length of the screen’s diagonal? Round to the nearest inch.

Find the values of the variables. Give your answers in simplest radical form.

18. 19. 20.

9.8 34° 21

RS = 6.8; RQ = 4.9

m∠GEF = 44°; distance from G to

−− DF = 3.7

XW = 174; BW = 59; BZ = 177

(-3, 4)PR = 51; GJ = 148; m∠GRP = 71°

∠E, ∠B, ∠H −− TY,

−− RY ,

−− RT

m∠SPV < m∠ZPV 2.5 < x < 8.5

triangle; obtuse

x = 10 √ � 2 x = 16; y = 16 √ � 3 x = 8√ � 3_3

;

y = 16√ � 3_3

102 ft 10 in.

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370 Chapter 5

C H A P T E R

5



Online Edition


Chapter 5 Tests





Answers 9. Possible answer:

Given: ∠1 and ∠2 form a lin. pair.Prove: ∠1 and ∠2 cannot both be

obtuse �.

Proof: Assume ∠1 and ∠2 are both obtuse �. By the def. of obtuse, m∠1 > 90° and m∠2 > 90°. If the 2 inequal-ities are added, m∠1 + m∠2 > 180°. However, by the Lin. Pair Thm., ∠1 and ∠2 are supp. By the def. of supp. �, this means that m∠1 + m∠2 = 180°. So m∠1 + m∠2 > 180° contradicts the given information. The assumption that ∠1 and ∠2 are both obtuse � is false. Therefore ∠1 and ∠2 cannot both be

obtuse.

12. greater than 133 mi and less than 361 mi

15. 3 √ �� 15 ; the side lengths do not form a Pythagorean triple because 3 √ �� 15 is not a whole number.

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Organizer N - mryildiz.weebly.commryildiz.weebly.com/uploads/1/7/2/0/17209270/chapter_1-5_answers.pdf · J is the midpoint of ... Items 9–11. Write the indicated type of statement