Chapter Test Form Amrwallmath.pbworks.com/w/file/fetch/90166925/chapter_test_ch._4... · Chapter...

Copyright © by Holt, Rinehart and Winston. 67 Holt GeometryAll rights reserved.

Name Date Class

CHAPTER

Circle the best answer.

1. What type of triangle is !ABC ?

30° 120°

30°

A acuteB equiangularC obtuseD right

2. How many sides must be congruent in an isosceles triangle?A at least 2B all 3

3. Which pair of angle measures CANNOT be the acute angles of a right triangle?A 29! and 61!

B 30! and 60!

C 38! and 53!

D 45! and 45!

4. What is m"ACD?

80°

50°

A 100B 130

5. Given: !TUV ! !TWV. What is the value of x ?

3 15

75

A 20B 123

6. If !KLM ! !RST, find the value of x.

A 18B 45

7. Given: "A ! "D, "B ! "E, "C ! "F,

_ AB !

_ DE , _

BC ! _

EF , and

_ CA !

_ FD . Which is a correct

congruence statement?A !BCA ! !DEFB !ABC ! !DEF

Use the figure for Exercises 8 and 9.

8. Which value for x proves that !ABC ! !DEF by SSS?A 7B 37

9. What additional information would allow you to prove the triangles congruent by SAS?A "E ! "DB "D ! "CC "D ! "AD "F ! "C

Chapter TestForm A4


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CHAPTER


10. Which postulate or theorem can you use to prove !ABE and !CDE ?A SSSB SASC ASAD AAS

11. What additional information will prove !ABE ! !CDE by HL?A _

AB ! _

CD B _

AE ! _

CE

12. To write a coordinate proof, you position a right isosceles triangle in the coordinate plane. The legs measure two units. What is the best position for the vertex angle?A (0,0)B (0,2)C (2, 0)D (2, 2)

13. Given: ABCD is a square with vertices A (0,0), B (0, 4), C (4, 4), and D (4,0). In a coordinate proof, what information would be used to prove

_ AB !

_ CD if you do

NOT use the distance formula?A x-coordinate of A, x-coordinate of CB y-coordinate of A, y-coordinate of CC y-coordinate of A, x-coordinate of CD x-coordinate of A, y-coordinate of C

14. What is the value of x?

A 22.5B 30C 45D 60


15. What postulate or theorem proves _

HG ! _

FG ?A Isosceles Triangle TheoremB Converse of Isosceles Triangle

Theorem

16. If !FGJ ! !HGJ, what reason justifies the statement "HGJ ! "FGJ?A ASAB Reflex. Prop. of !C Def. of bisectsD CPCTC

Chapter TestForm A continued4


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Chapter TestForm B

CHAPTER


1. Classify the triangle.

60°85°

35°

A isosceles acute C scalene acuteB isosceles obtuse D scalene obtuse


60° 60°

60°3 1 5

2. Which is NOT a correct classification for the triangle?F acute H isoscelesG equiangular J scalene

3. What is the length of side _

BC ?A 3 C 10B 8 D 24


(8 18)°

(6 4)°

(20 4)°

4. What is m"KLM?F 3 H 42G 22 J 64

5. What is m"M?A 0.2 C 26B 4 D 64

6. What is the m"U?

F 5 H 40G 15 J 120

7. Two congruent triangles have the following corresponding parts: _

RS ! _

UV , _

RT ! _

UW , and "R ! "U. Which is NOT necessarily a correct congruence statement?A !RST ! !UVWB !STR ! !VWUC !TRS ! !VWUD !TRS ! !WUV

8. !KLM ! !RST. m"L " (3x # 15)! and m"S " (6x # 3)!. What is the value of x?F 2 H 6G 4 J 27

Use the figure for Exercises 9 –12.

9. If AD " 5y # 7 and BC " 7y $ 3, what must the value of y be to prove !AED ! !CEB by the SSS Postulate?A 2 C 17B 5 D 32

10. What postulate or theorem justifies the congruence statement !ABE ! !CDE?F SSS H ASAG SAS J AAS

4


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CHAPTER Chapter TestForm B continued

11. If "B and "C are right angles, what additional congruence statement would allow you to prove !DCB ! !ABC by the ASA postulate?A "DBC ! "ACBB "BDC ! "CABC _

AB ! _

DC D _

AC ! _

DB

12. If "A and "C are right angles and _

AD ! _

BC , what postulate or theorem justifies the congruence statement !BCD ! !DAB?F SAS H AASG ASA J HL

13. A right triangle with leg lengths of 4 and 3 units has to be positioned in the coordinate plane to write a coordinate proof. Which set of coordinates would make the proof easier to complete?A (4, 0), (0, 0), (4, 3)B (3, 0), (0, 0), ($4, 0)C (0, 4), (0, 0), ($3, 0)D (0, 4), (0, 0), (3, 0)

14. Which of the following would you find most useful in giving a coordinate proof that two triangles are congruent by SSS?F Distance Formula H CPCTCG Midpoint Formula J Slope Formula


(4 12)°

A 12 C 18B 19.5 D 60

Use the partially completed two-column proof for Exercises 16–18.

Given: _

GJ bisects "FGH, _

FG ! _

HG

Prove: _

FJ ! _

HJ Proof:

Statements Reasons

1. _

GJ bisects "FGH. 1. Given

2. "FGJ ! "HGJ 2. Def. of " bisector

3. _

FG ! _

HG 3. Given

4. "F ! "H 4. ?

5. !FGJ ! !HGJ 5. ?

6. _

FJ ! _

HJ 6. ?

16. Which reason belongs in Step 4?F Isosc. ! Thm.G Conv. of Isosc. ! Thm.H ASAJ Def. of " bisector

17. Which reason belongs in Step 5?A Isosc. ! Thm. C CPCTCB ASA D HL

18. Which reason belongs in Step 6?F Isosc. ! Thm.G ASAH CPCTCJ Def. of " bisector

4


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CHAPTER


1. Which best describes !ABC with vertices A($2, 1), B (0, 4), and C (2, 1)?A acute C obtuseB equiangular D right

2. Which is a correct classification of !DEF with vertices D ($3, $2), E ($2, 3), and F (1, 0)?F equilateral H scaleneG isosceles J Not here

3. If the acute angles of a right triangle are congruent, which statement is NOT true?A Both acute angles measure 45!.B Only one exterior " measures 90!.C Only one exterior " measures 135!.D Two exterior angles measure 135!.


°

122°

139°

F 41 H 99G 58 J 122

5. !QRS ! !STQ, QS " x 2 $ 10 and SQ " $2x $ 2. What is the value of x ?A $4 C 2B $2 D 4

6. !ABC ! !DEF. What information is NOT needed to find the perimeter of !ABC if you are given all four lengths below?

F DE H CFG BG J EF

Use the partially completed two-column proof for Exercise 7.

Given:

Prove: !GHF ! !MOL

Proof:

Statements Reasons1. _

GF ! _

ML , _

FH ! _

LO , _

GH ! _

MO

1. Given

2. "F ! "L 2. ?

3. "H ! "O 3. Given

4. "G ! "M 4. ?

5. !GHF ! !MOL 5. ?

7. Which reason does NOT belong in the proof?A Def. of ! !sB Third # Thm.C Rt. " ! Thm.D CPCTC

Use the figure for Exercises 8–11.

8. AB " y 2 # 3, DC " 3y # 1, EB " 3y $ 1, ED " y 2 # 1, AE " y 2, CE " 2y. What value of y proves !AEB ! !CED by the SSS Postulate?F $2 H 1G $1 J 2

Chapter TestForm C4


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CHAPTER

9. What information would allow you to prove !AED ! !CEB by SAS?A E is the midpoint of

_ DB .

B E is the midpoint of _

AC .C E bisects

_ AC .

D E bisects both _

DB and _

AC .

10. If "ADC and "ABC are right angles, AC " BD, and AB " DC, which postulate or theorem proves !ABC ! !CDA?F SSS H ASAG SAS J HL

11. If _

AD " _

BC and "ABD ! "CDB, which postulate or theorem could be used to prove !ABD ! !CDB?A SAS C SSSB ASA D HL

12. Given: !ABC with vertices A (5, $2), B (5, $7), and C (1, $2). Which set of coordinates best repositions the triangle to make a coordinate proof easier?F (0, 0), (4, 0), and (4, $5)G (0, 0), ($4, 0), and (0, 5)H (0, 0), (0, 4), and (5, 0)J (0, 0), (4, 0), and (0, $5)

13. If the height of a right triangle is n units and the base is m units, which statement is NOT true?

A The midpoint of the hypotenuse is (m, n).

B A (0, 0), B (m, 0), and C (0, n ) can represent the vertices.

C The midpoint of the hypotenuse is # m __ 2 , n __ 2 $ .

D The slope of the hypotenuse is $ n __ m .

14. What is m"DAC?

F 30! H 60!

G 45! J Not here

Use the partially completed two-column proof for Exercises 15 and 16.

Given: _

JK ! _

LK ; "JYL and "LXJ are rt. #.

Prove: _

JY ! _

LX

Proof:

Statements Reasons

1. "KJL ! "KLJ 1. ?

2. _

JL ! _

LJ 2. ?

3. "JYL and "LXJ are rt. #.

3. Given

4. "JYL ! "LXJ 4. ?

5. !JYL ! !LXJ 5. ?

6. _

JY ! _

LX 6. ?

15. Which justification belongs in Step 1?A Isosc. ! Thm. C Rt. " ! Thm.B Reflex. Prop. of ! D CPCTC

16. Which justification belongs in Step 6?F Isosc. ! Thm. H Rt. " ! Thm.G HL J CPCTC

Chapter TestForm C continued4


Name Date Class

CHAPTER Chapter TestForm A


80°

50° 50°

1. Classify the triangle by its angle measures.

2. Classify the triangle by its side lengths.

3. Complete the sentence. All of the angles in an equilateral triangle measure .

4. What is the measure of "1?

45°

1

5. Given: !GHJ ! !NOP. What is the value of x ?

35° °

6. If !KLM ! !RST, what is the value of x ?

132 3

7. Complete the statement. Two triangles are congruent if and only if their

angles and sides are congruent.


4 327

8. What value of x proves !ABC ! !DEF by SAS?

9. If _

AB ! _

DE , what additional congruence statement is needed to prove !ABC ! !DEF by SSS?


10. Write True or False. You can use AAS to prove !ABE ! !CDE.

11. What additional congruence statement is needed to prove !ABE ! !CDE by HL?

4


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CHAPTER

12. Write True or False. To give a coordinate proof about a right triangle, it is a good idea to position the vertex of the right angle at (0, 0) so that the legs are on the positive sides of the axes.

13. Write True or False. The Midpoint Formula is used in the coordinate proof to prove the statement

EF " 1 __ 2 RS.

14. Find the value of x.

74°

( 12)°

15. What postulate or theorem proves "K ! "M ?

16. Write True or False. Given: !ABC and !DEF. To use CPCTC to prove "A ! "D, you must first prove !ABC ! !DEF.

Chapter TestForm A continued4


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CHAPTER Chapter TestForm B


1. Classify !ABD by its angle measures.

right 2. Classify !ABC by its side lengths.

equilateral 3. The measure of the smallest angle of a

right triangle is 27!. What is the measure of the second to smallest angle?

63!

4. Find the measure of "RST.

(5 4)°

(8 4)°

52!

5. !JKL ! !MNP, KL " 21x $ 2, NP " 20x, LJ " 15x, PM " 13x # 4. Find LJ.

30 6. Given: !TUV ! !TWV. Find m"U

and UV.

(9 6)°

(7 22)°

6 5

5 7

m!U " 120!; UV " 17

7. Given: "5 ! "6, "3 ! "4, _

DE ! _

FE , _

FG ! _

DG , _

GE ! _

GE . Provide an additional statement and a reason for that statement to prove !DEG ! !FEG by the definition of congruent triangles.

15

6

2

34

!1 ! !2; Third " Thm.


8. If AB " 3x $ 7 and DC " 2x # 1, what value of x proves !AEB ! !CED by the SSS Postulate?

x " 8 9. What postulate or theorem proves

!AED ! !CEB?

SAS 10. If "DAB ! "ADC, what additional

congruence statement do you need to prove !DAB ! !ADC by the ASA Postulate?

!ADB ! !DAC

11. If "ABC and "CDA are right angles and _

AB ! _

CD , what postulate or theorem proves !ABC ! !CDA?

HL

4


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CHAPTER

Use the Given information for Exercises 12 and 13.

Given: An isosceles triangle ABC with _

AB ! _

BC and a perpendicular bisector

_ BD from B to

_ AC .

12. Position the figure in the coordinate plane and assign coordinates to each point so proving that the area of !ABD is equal to the area of !CBD using a coordinate proof would be easier to complete. Possible answer:

A(#2, 0), B(0, 4), C(2, 0), D(0,0) 13. Write a coordinate proof to prove that

the area of !ABD is equal to the area of !CBD. Possible answer:

#ABD is a right triangle with

base AD and height BD. #CBD

is a right triangle with base CD

and height BD.

area of #ABD "

1 __ 2 bh " 1 __

2 (4)(2) " 4 square units

area of #CBD "

1 __ 2 bh " 1 __

2 (4)(2) " 4 square units

The area of #ABD is equal to

4 square units, which equals the

area of #CBD.

14. Find the value of x.

(3 5)°

2 °

x " 25

Use the figure and the partially completed two-column proof for Exercises 15 and 16.

Given: "BAC ! "BCA

Prove: _

AD ! _

CE

Proof:

Statements Reasons1. "BAC ! "BCA 1. Given

2. _

BA ! _

BC 2. ?

3. "D and "E are right #.

3. Given (diagram)

4. _

DB ! _

EB 4. Given (diagram)

5. !DBA ! !EBC 5. HL Congruence Thm.

6. _

AD ! _

CE 6. ?

15. What is the justification for Step 2?

Conv. of Isosc. # Thm. 16. What is the justification for Step 6?

CPCTC

Chapter TestForm B continued4


Name Date Class

CHAPTER Chapter TestForm C


60°60°

60°

30°

1. Classify !ABC by angle measures.

2. Classify !ABD by side lengths.


(2 10)° (2 10)°

3 °

3. What is m"T ?

4. What is the value of y ?

5. Given !QRS ! !STQ, "R " 4x 2 $ 4, and "T " 3x 2 $ 3x. What is m"R?

6. Given !QRS ! !STQ, RS " 3x $ 3, TQ " 2x # 2, and QR " x 2 $ 2. What is the length of side

_ ST ?

7. Prove !TUV ! !TWV by using the definition of congruent triangles.


8. If _

AD ! _

BC , write a statement about point E that would allow you to prove !AED ! !CEB by the SSS Postulate.

9. Suppose _

AE ! _

CE and _

BE ! _

DE . What postulate or theorem will allow you to prove !BEA ! !DEC ?

10. Write True or False. If "ABC and "DCB are right angles and

_ AD " _

BC , you can prove !ABC ! !DCB.

11. "DAB and "BCD are right angles. Write a single congruence statement about two segments that would allow you to conclude that !DAB ! !BCD. What theorem or postulate would justify the conclusion?

4


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CHAPTER Chapter TestForm C continued

12. A triangle has vertices P(a, b), Q(c, d ), and R (e, f ). You are asked to prove that the image !P %Q %R % of !PQR after reflection across the y-axis is congruent to the preimage. What coordinates should you use for the vertices of !P %Q %R %?

13. Assign variables as the coordinates and write a coordinate proof.

Given: Square ABCD with side length of d units

Prove: AC " BD

14. What is the value of x ?

Use the figure and the partially completed two-column proof for Exercises 15 and 16.

Given: _

RU ! _

TV , _

RS ! _

TS

Prove: _

RV ! _

TU

Proof:

Statements Reasons

1. _

RS ! _

TS 1. Given

2. "SRT ! "STR 2. ?

3. m"SRT " m"STR 3. Def. of ! #

4. m"RTV " 180! $ m"STR

4. Lin. Pair Thm.

5. m"TRU " 180! $ m"STR

5. Lin. Pair Thm. and Subst. (Step 2)

6. m"RTV " m"TRU 6. Subst. Prop. of "

7. "RTV ! "TRU 7. Def. of ! #

8. _

RT ! _

RT 8. Reflex. Prop.

of !

9. _

RU ! _

TV 9. Given

10. !RTV ! !TRU 10. ?

11. _

RV ! _

UT 11. ?

15. What reason belongs in Step 2?

16. What reason belongs in Step 11?

4


Answer Key continued

Section Quiz: Lessons 4-4 Through 4-8

1. B 6. G

2. J 7. C

3. D 8. H

4. J 9. D

5. D 10. H

Chapter Test Form A: Multiple Choice

1. C 9. D

2. A 10. C

3. C 11. B

4. B 12. A

5. A 13. B

6. A 14. B

7. B 15. B

8. A 16. D

Chapter Test Form B: Multiple Choice

1. C 7. C

8. G

9. B

10. G

11. A

12. J

13. D

14. F

15. A

16. F

17. B

18. H

2. J

3. B

4. J

5. C

6. J

Chapter Test Form C: Multiple Choice

1. A 9. D

2. H 10. J

3. C 11. B

4. H 12. H

5. A 13. A

6. G 14. F

7. D 15. A

8. J 16. J

Chapter Test Form A: Free Response

1. acute

2. isosceles

3. 60!

4. 45!

5. 35

6. 5

7. corresponding

8. 6

9. _

AC ! _

DF

10. True

11. _

EC ! _

EA

12. True

13. False

14. 62

15. Isosceles Triangle Theorem

16. True

Chapter Test Form B: Free Response

1. right

2. equilateral

3. 63!

4. 52!

5. 30

6. m!U " 120!; UV " 17

7. !1 ! !2; Third " Thm.

8. 8

9. SAS

10. !ADB ! !DAC

11. HL

12. Possible answer: A (#2, 0), B (0, 4), C (2, 0), D(0,0)


Answer Key continued

13. Possible answer: !ABD is a right triangle

with base AD and height BD. !CBD is

a right triangle with base CD and height

BD. Area of !ABD ! 1 __ 2 bh ! 1 __ 2 (4)(2) !

4 square units. Area of !CBD ! 1 __ 2 bh

! 1 __ 2

(4)(2) ! 4 square units. The area of

!ABD is equal to 4 square units, which

equals the area of !CBD.

14. x ! 25

15. Conv. of Isosc. ! Thm.

16. CPCTC

Chapter Test Form C: Free Response

1. equiangular, acute

2. scalene

3. 60"

4. y ! 80

5. 60"

6. 23 units

7.

Statements Reasons1. _

TU ! _

TW , _

UV ! _

WV 1. Given

2. _

VT ! _

VT 2. Reflex. Prop. of !3. "VTU ! "VTW,

"UVT ! "WVT3. Given

4. "U ! "W 4. Third # Thm.5. !TUV ! !TWV 5. Def. of ! !s

8. E bisects both _

AC and _

DB , or E is the midpoint of both

_ AC and

_ DB .

9. SAS

10. True

11. _

AD ! _

BC or _

AB ! _

DC ; HL

12. P #($a, b ), Q #($c, d ), and R #($e, f )

13. Possible answer: Use vertices A (0, 0), B (0, d ), C (d, d ), and D (d, 0), where d % 0.

AC ! "!

(d $ 0)2 & (d $ 0)2 !

"!

2d 2 ! d "! 2

BD ! "!

(d $ 0)2 & (0 $ d )2 !

"!

2d 2 ! d "! 2

AC ! BD by the Subst. Prop. of !

14. 48

15. Isosceles Triangle Theorem

16. CPCTC

Performance Assessment

1. B (8, 5)

2. Def. of midpoint

3. ‹ __

› AE : 4 __ 3 ;

‹ __

› EB :

$ 3 __ 4 ;

‹ __

› BD :

$ 3 __ 4 ;

‹

___ › DC : 4 __ 3

4. Yes; since 4 __ 3 ! $ 3 __ 4 ! $1,

‹ __

› AE $

‹ __

› EB , so

m"AEB ! 90". Also, since $

3 __ 4

! 4 __ 3 ! $1,

‹ __

› BD $

‹

___ › DC , so m"CDB ! 90".

5. Possible answer: Show that _

AE ! _

CD or _

EB ! _

DB by using the Distance Formula.

6. CPCTC

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