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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All rights reserved.
258
Holt Geometry