Name CHAPTER
4
Date
Class
Chapter Test Form A 6. If !KLM ! !RST, find the value of x.
Circle the best answer.
+
1. What type of triangle is !ABC ?
2
X
! 30°
$
A B C D
30°
120°
,
"
4
3
A 18 B 45
acute equiangular obtuse right
7. Given: "A ! "B _ ! "E,_ _"D, _ AB ! DE , BC ! EF , "C ! "F, _ _ and CA ! FD . Which is a correct congruence statement? A !BCA ! !DEF B !ABC ! !DEF
2. How many sides must be congruent in an isosceles triangle? A at least 2 B 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!
Use the figure for Exercises 8 and 9. &
X
$ "
!
50°
$
A 100 B 130
9. What additional information would allow you to prove the triangles congruent by SAS? A "E ! "D B "D ! "C C "D ! "A D "F ! "C
5. Given: !TUV ! !TWV. What is the value of x ? 5 3X 15
4
#
8. Which value for x proves that !ABC ! !DEF by SSS? A 7 B 37
80°
#
%
!
4. What is m"ACD ?
"
-
6 75
7
A 20 B 123 Copyright © by Holt, Rinehart and Winston. All rights reserved.
67
Holt Geometry
Name CHAPTER
4
Date
Class
Chapter Test Form A continued 14. What is the value of x?
Use the figure for Exercises 10 and 11. " # % ! $
X
10. Which postulate or theorem can you use to prove !ABE and !CDE ? A SSS
A B C D
B SAS C ASA D AAS
Use the figure for Exercises 15 and 16.
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)
*
&
'
15. What or theorem proves _ postulate _ HG ! FG ? A Isosceles Triangle Theorem B Converse of Isosceles Triangle Theorem 16. If !FGJ ! !HGJ, what reason justifies the statement "HGJ ! "FGJ ? A ASA B Reflex. Prop. of ! C Def. of bisects D CPCTC
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 would _ _ information be used to prove AB ! CD if you do NOT use the distance formula? A x-coordinate of A, x-coordinate of C B y-coordinate of A, y-coordinate of C C y-coordinate of A, x-coordinate of C D x-coordinate of A, y-coordinate of C
Copyright © by Holt, Rinehart and Winston. All rights reserved.
22.5 30 45 60
68
Holt Geometry
Name
Date
Class
Chapter Test
CHAPTER
4
Form B
Circle the best answer.
6. What is the m"U ?
1. Classify the triangle.
X
4
85° 60°
6 X
35°
"
!
60°
3X 1 60°
#
2. Which is NOT a correct classification for the triangle? F acute H isosceles G equiangular J scalene
8. !KLM ! !RST. m"L " (3x # 15)! and m"S " (6x # 3)!. What is the value of x? F 2 H 6 G 4 J 27
_
3. What is the length of side BC ? A 3 C 10 B 8 D 24
Use the figure for Exercises 9 –12.
Use the figure for Exercises 4 and 5. +
, *
!
$
(20X 4)°
(6X 4)°
4. What is m"KLM? F 3 G 22
H 42 J 64
5. What is m"M? A 0.2 B 4
C 26 D 64
Copyright © by Holt, Rinehart and Winston. All rights reserved.
" %
-
(8X 18)°
H 40 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 ! !UVW B !STR ! !VWU C !TRS ! !VWU D !TRS ! !WUV
Use the figure for Exercises 2 and 3. 60°
7
F 5 G 15
A isosceles acute C scalene acute B isosceles obtuse D scalene obtuse
X5
5
#
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 17 B 5 D 32 10. What postulate or theorem justifies the congruence statement !ABE ! !CDE ? F SSS H ASA G SAS J AAS
69
Holt Geometry
Name CHAPTER
4
Date
Class
Chapter Test Form B
continued Use the partially completed two-column proof for Exercises 16–18.
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 ! "ACB B "BDC _ ! "CAB _ ! DC C AB _ _ D AC ! DB
_
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 ASA J Def. of " bisector 17. Which reason belongs in Step 5? A Isosc. ! Thm. C CPCTC B ASA D HL 18. Which reason belongs in Step 6? F Isosc. ! Thm. G ASA H CPCTC J Def. of " bisector
(4X 12)°
Copyright © by Holt, Rinehart and Winston. All rights reserved.
_
_
15. What is the value of x?
D 60
&
Prove: FJ ! HJ Proof:
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 CPCTC G Midpoint Formula J Slope Formula
B 19.5
_
'
_
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)
C 18
*
(
12. If_ "A and _ "C are right angles and AD ! BC , what postulate or theorem justifies the congruence statement !BCD ! !DAB? F SAS H AAS G ASA J HL
A 12
_
Given: GJ bisects "FGH, FG ! HG
70
Holt Geometry
Name
Date
CHAPTER
4
Class
Chapter Test Form C Use the partially completed two-column proof for Exercise 7.
Circle the best answer. 1. Which best describes !ABC with vertices A($2, 1), B(0, 4), and C (2, 1)? A acute C obtuse B equiangular D right
Given: '
2. Which is a correct classification of !DEF with vertices D($3, $2), E ($2, 3), and F(1, 0)? F equilateral H scalene G isosceles J Not here
&
-
(
,
/
Prove: !GHF ! !MOL Proof:
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!.
!_ ML, 1. GF _ FH ! LO , _ _ GH ! MO
1. Given
2. "F ! "L
2.
4. What is the value of x?
3. "H ! "O
3. Given
4. "G ! "M
4.
?
5. !GHF ! !MOL
5.
?
Statements
_
X° 139° 122°
F 41 G 58
2 5. !QRS ! !STQ, QS " x $ 10 and SQ " $2x $ 2. What is the value of x ? A $4 C 2 B $2 D 4
!
$
&
F DE G BG Copyright © by Holt, Rinehart and Winston. All rights reserved.
#
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 1
' #
" %
%
$
?
Use the figure for Exercises 8–11.
6. !ABC ! !DEF. What information is NOT needed to find the perimeter of !ABC if you are given all four lengths below?
!
Reasons
7. Which reason does NOT belong in the proof? A Def. of ! !s B Third # Thm. C Rt. " ! Thm. D CPCTC
H 99 J 122
"
_
H CF J EF
G $1
71
J 2 Holt Geometry
Name
Date
Class
Chapter Test
CHAPTER
4
Form C
continued
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 .
14. What is m"DAC? !
$
F 30! G 45!
10. If "ADC and "ABC are right angles, AC " BD, and AB " DC, which postulate or theorem proves !ABC ! !CDA? F SSS H ASA G SAS J HL _
_
_
_
Given: JK ! LK ; "JYL and "LXJ are rt. #. +
8
9
*
,
_
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)
_
Prove: JY ! LX Proof:
Statements
?
2. JL ! LJ
2.
?
3. "JYL and "LXJ are rt. #.
3. Given
4. "JYL ! "LXJ
4.
?
5. !JYL ! !LXJ
5.
?
6. JY ! LX
6.
?
_
A The midpoint of the hypotenuse is (m, n).
_
B A(0, 0), B(m, 0), and C(0, n ) can represent the vertices.
Reasons 1.
1. "KJL ! "KLJ
13. If the height of a right triangle is n units and the base is m units, which statement is NOT true?
_
_
15. Which justification belongs in Step 1? A Isosc. ! Thm. C Rt. " ! Thm. B Reflex. Prop. of ! D CPCTC
C The midpoint of the hypotenuse is m, __ n . __ 2 2
$
16. Which justification belongs in Step 6? F Isosc. ! Thm. H Rt. " ! Thm. G HL J CPCTC
n D The slope of the hypotenuse is $ __ m.
Copyright © by Holt, Rinehart and Winston. All rights reserved.
H 60! J Not here
Use the partially completed two-column proof for Exercises 15 and 16.
11. If AD " BC and "ABD ! "CDB, which postulate or theorem could be used to prove !ABD ! !CDB? A SAS C SSS B ASA D HL
#
"
#
72
Holt Geometry
Name
Date
Class
Chapter Test
CHAPTER
4
Form A
Use the figure for Exercises 1 and 2.
7. Complete the statement. Two triangles are congruent if and only if their angles and sides are congruent.
80°
50°
50°
1. Classify the triangle by its angle measures.
Use the figure for Exercises 8 and 9. &
2. Classify the triangle by its side lengths.
%
!
27
4X 3
$
"
3. Complete the sentence. All of the angles in . an equilateral triangle measure
#
8. What value of x proves !ABC ! !DEF by SAS?
4. What is the measure of "1? _
1
45°
5. Given: !GHJ ! !NOP. What is the value of x ? '
0
Use the figure for Exercises 10 and 11. "
.
# !
*
_
9. If AB ! DE , what additional congruence statement is needed to prove !ABC ! !DEF by SSS?
$
X°
35°
(
%
/
10. Write True or False. You can use AAS to prove !ABE ! !CDE. 6. If !KLM ! !RST, what is the value of x ? +
2
2X 3
,
11. What additional congruence statement is needed to prove !ABE ! !CDE by HL?
13
-
4
Copyright © by Holt, Rinehart and Winston. All rights reserved.
3
73
Holt Geometry
Name
Date
Class
Chapter Test
CHAPTER
4
Form A continued
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.
15. What postulate or theorem proves "K ! "M ? +
,
-
.
13. Write True or False. The Midpoint Formula is used in the coordinate proof to prove the statement 1 RS. EF " __ 2
16. Write True or False. Given: !ABC and !DEF. To use CPCTC to prove "A ! "D, you must first prove !ABC ! !DEF.
Y
2 %
$
X
&
3
14. Find the value of x. (X 12)°
74°
Copyright © by Holt, Rinehart and Winston. All rights reserved.
74
Holt Geometry
Name
Date
CHAPTER
4
Class
Chapter Test Form B
_
!
$
#
"
$ 6
1. Classify !ABD by its angle measures.
' 3 4
right
&
2. Classify !ABC by its side lengths.
!
$
4. Find the measure of "RST. 2
4
9. What postulate or theorem proves !AED ! !CEB?
5. !JKL ! !MNP, KL " 21x $ 2, NP " 20x, LJ " 15x, PM " 13x # 4. Find LJ.
SAS
30
10. If "DAB ! "ADC, what additional congruence statement do you need to prove !DAB ! !ADC by the ASA Postulate?
6. Given: !TUV ! !TWV. Find m"U and UV.
(7X 22)°
5
!ADB ! !DAC
6Y 5
6
11. If_ "ABC and "CDA are right angles and _ AB ! CD , what postulate or theorem proves !ABC ! !CDA?
5Y 7
7
m!U " 120!; UV " 17
Copyright © by Holt, Rinehart and Winston. All rights reserved.
#
x"8
52!
4
"
8. If AB " 3x $ 7 and DC " 2x # 1, what value of x proves !AEB ! !CED by the SSS Postulate?
(8X 4)°
(9X 6)°
%
%
63!
3
2 1
Use the figure for Exercises 8–11.
3. The measure of the smallest angle of a right triangle is 27!. What is the measure of the second to smallest angle?
1
5
!1 ! !2; Third " Thm.
equilateral
(5X 4)°
_
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.
Use the figure for Exercises 1 and 2.
HL
75
Holt Geometry
Name CHAPTER
4
Date
Class
Chapter Test Form B
continued 14. Find the value of x.
Use the Given information for Exercises 12 and 13.
(3X 5)°
Given: An triangle ABC with _ _ isosceles AB ! BC_ and a perpendicular _ bisector BD from B to AC .
2X °
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:
x " 25 Use the figure and the partially completed two-column proof for Exercises 15 and 16. Given: "BAC ! "BCA
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
#
_
_
Prove: AD ! CE
base AD and height BD. #CBD
Proof: Statements
is a right triangle with base CD
1. "BAC ! "BCA
and height BD.
_
area of #ABD " 1(4)(2) " 4 square units 1bh " __ __ 2 2
_
Reasons 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.
_
area of #CBD " 1(4)(2) " 4 square units 1bh " __ __ 2 2
_
_
_
6. AD ! CE
?
6.
15. What is the justification for Step 2?
The area of #ABD is equal to
Conv. of Isosc. # Thm.
4 square units, which equals the
16. What is the justification for Step 6?
CPCTC
area of #CBD.
Copyright © by Holt, Rinehart and Winston. All rights reserved.
%
76
Holt Geometry
Name
Date
Class
Chapter Test
CHAPTER
4
Form C 7. Prove !TUV ! !TWV by using the definition of congruent triangles.
Use the figure for Exercises 1 and 2. !
5
60° 30°
$
60°
60°
#
4
"
6 7
1. Classify !ABC by angle measures.
2. Classify !ABD by side lengths.
Use the figure for Exercises 3 and 4. 4
Use the figure for Exercises 8–11.
3X °
! 6
(2X 10)°
" %
(2Y 10)°
5
7 $
3. What is m"T ?
#
_
_
8. If AD ! BC , write a statement about point E that would allow you to prove !AED ! !CEB by the SSS Postulate.
4. What is the value of y ?
_
_
_
_
9. Suppose AE ! CE and BE ! DE . What postulate or theorem will allow you to prove !BEA ! !DEC ?
5. Given !QRS ! !STQ, "R " 4x 2 $ 4, 2 and "T " 3x $ 3x. What is m"R?
6. Given !QRS ! !STQ, RS " 3x $ 3, 2 TQ " 2x # 2, and QR _" x $ 2. What is the length of side ST ?
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?
Copyright © by Holt, Rinehart and Winston. All rights reserved.
77
Holt Geometry
Name CHAPTER
4
Date
Class
Chapter Test Form 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%?
Use the figure and the partially completed two-column proof for Exercises 15 and 16. _ _
_
_
Given: RU ! TV , RS ! TS 3
2
4
5
13. Assign variables as the coordinates and write a coordinate proof.
6
_
_
Prove: RV ! TU
Given: Square ABCD with side length of d units
Proof: Statements
Prove: AC " BD
_
_
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 14. What is the value of x ?
9. RU ! TV X
Reasons ?
8. Reflex. Prop. of ! 9. Given
10. !RTV ! !TRU
10.
?
11. RV ! UT
11.
?
_
_
15. What reason belongs in Step 2?
Copyright © by Holt, Rinehart and Winston. All rights reserved.
16. What reason belongs in Step 11?
78
Holt Geometry
Answer Key
continued
Section Quiz: Lessons 4-4 Through 4-8
Chapter Test Form A: Free Response
1. B
6. G
1. acute
2. J
7. C
2. isosceles
3. D
8. H
3. 60!
4. J
9. D
4. 45!
5. D
10. H
5. 35 6. 5
Chapter Test Form A: Multiple Choice
7. corresponding
1. C
9. D
2. A
10. C
8. 6
3. C
11. B
9. AC ! DF
4. B
12. A
10. True
5. A
13. B
11. EC ! EA
6. A
14. B
12. True
7. B
15. B
13. False
8. A
16. D
14. 62
_
_
Chapter Test Form B: Multiple Choice
_
_
15. Isosceles Triangle Theorem 16. True
1. C
7. C
13. D
2. J
8. G
14. F
3. B
9. B
15. A
1. right
4. J
10. G
16. F
2. equilateral
5. C
11. A
17. B
3. 63!
6. J
12. J
18. H
4. 52!
Chapter Test Form B: Free Response
5. 30
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
Copyright © by Holt, Rinehart and Winston. All rights reserved.
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)
257
Holt Geometry
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 1 bh ! __ 1 (4)(2) ! BD. Area of !ABD ! __ 2 2 1 bh 4 square units. Area of !CBD ! __ 2 1(4)(2) ! 4 square units. The area of ! __ 2 !ABD is equal to 4 square units, which
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
equals the area of !CBD.
AC ! BD by the Subst. Prop. of !
14. x ! 25
14. 48
15. Conv. of Isosc. ! Thm.
15. Isosceles Triangle Theorem
16. CPCTC
16. CPCTC
Chapter Test Form C: Free Response
Performance Assessment
1. equiangular, acute
1. B (8, 5)
2. scalene
2. Def. of midpoint __
3. 60" 4. y ! 80 5. 60" 7. Statements _
_
_
Reasons
! TW , 1. TU _ _ UV ! WV
1. Given
2. VT ! VT
2. Reflex. Prop. of !
3. "VTU ! "VTW, "UVT ! "WVT
3. Given
4. "U ! "W
4. Third # Thm.
5. !TUV ! !TWV
_
__
__
3 ! $1, ‹AE› $ ‹EB›, so 4 __ 4. Yes; since __ ! 3 $4 3 __ 4 ! $1, m"AEB ! 90". Also, since __ $4 ! 3 ‹___› ‹__› BD $ DC, so m"CDB ! 90".
6. 23 units
_
__
‹ › 4 ‹ › 3; ; EB: __ 3. AE: __ 3 $4 ___ __ ‹ › 3 ; ‹DC›: __ 4 BD: __ 3 $4
_
_
5. Possible answer: Show that AE ! CD _ _ or EB ! DB by using the Distance Formula. 6. CPCTC
5. Def. of ! !s _
8. E bisects both AC and DB , or E is the _ _ midpoint of both AC and DB . 9. SAS 10. True _
_
_
_
11. AD ! BC or AB ! DC ; HL 12. P #($a, b ), Q #($c, d ), and R #($e, f )
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258
Holt Geometry
