Geometry
Chapter 4
Homework
Packet
Name_____________
Period____________
Teacher___________
Special thanks to Mrs. Gerardot for sharing.
Geometry HW
Sections 4.1 and 4.2
Name____________________________________#_________
ALL ANSWERS ARE POSTED ON-LINE AND IN THE CLASSROOM.
BE SURE TO CHECK THEM BEFORE CLASS STARTS.
Draw and label the following diagrams.
1.
Isosceles triangle, MNT , with
vertex angle M.
2.
Right triangle, DOG , with
right angle G.
Find the measure of each numbered angle. You MUST show your work even if you did it mentally.
3.
4.
#1
#1
5.
◦
56
◦
40
#3
#2
◦
42
◦
45
#3
#1
◦
45
#2
Triangle Sum Theorem: The variable expressions represent the angle measures of a triangle. Solve for x
then find the measure of each angle then classify the triangle by its angles. You must show all work for
credit.
6.
mA  x
7.
mR  x
8.
mW   x  15 
mB   2 x 
mS   7 x 
mY   2 x  165 
mC   2 x  15 
mT  x
mZ  90
Solve for x:
Solve for x:
Solve for x:
Classify by its angles…
Classify by its angles…
Classify by its angles…
right acute obtuse equiangular
right acute obtuse equiangular
right acute obtuse equiangular
Exterior Angle Theorem: Solve for x in each diagram then find the measure of each exterior angle shown.
(You do not have to draw in additional exterior angles.) You must show all work for credit.
9.
10.
11.
◦
38
◦
(2x - 8)
◦
x
◦
(7x + 1)
◦
31
◦
y
(10x + 9)
◦
◦
x
◦
(2x - 21)
Solve for x:
Solve for x:
Solve for x:
Exterior Angle = _________
Exterior Angle = _________
Exterior Angle = _________
Congruent Figures:
Complete each statement and justify each answer with a theorem, definition, or
postulate.
Given: ABC  TUZ
U
B
◦
8 cm 59
A
Z
55◦
C
T
Hint: What does it mean when two triangles are congruent? Mark the corresponding parts based on the congruence
statement BEFORE answering the questions below.
12.
B  ________
14.
Why?________________
BC  ________
(Hint: What does”no bar”mean?
What should your answer look like?)
Why?________________
13.
ZT  ________
15.
mA  m ________  _______
Why?________________
Why?________________
Proof Sequences: Use the given information to complete the geometric statements. Some given has been
marked in the diagram for you and some has not. Explain each answer with a theorem, definition, or
postulate.
16.
All given information has been marked.
A
D
B
2
1
All given information has been marked.
17.
G
C
F
1
H
2
J
K
1  ________ Why?__________________
BD  ________ Why?__________________
1  ________ Why?__________________
ABD  ________ Why?__________________
FGH  ________ Why?__________________
18.
Given: C is the midpoint of BD
ABC is isosceles
Given: WZ  YX
WZ is an angle bisector
19.
A
W
3 4
Y
D
C
1
2
Z
X
B
AD  ________ Why?__________________
1 & 2 are ______________ Why?__________
DC  ________ Why?__________________
1  ________ Why?__________________
AC  ________ Why?__________________
3  ________ Why?__________________
BCA  ________ Why?__________________
WZ  ________ Why?__________________
DAC  ________ Why?__________________
WZY  ________ Why?__________________
Review: Find the indicated values. Simplify any radicals. You must show all work for credit.
20.
Find AB
A(3, 8) and B(-1, -4)
21.
Find CD
C (0, -4) and D (9, 2)
22.
Find the midpoint
(-7, 5) and (-3, -9)
23.
Find the midpoint
(1, -7) and (-6, 4)
24.
Find the other endpoint
Endpoint (-7, 5) and Midpoint (-3, -9)
25.
Find the other endpoint
Endpoint (1, -7) and Midpoint (-6, 4)
26.
The measure of an angle is 30 less than
5 times its supplement. Find both angles.
27.
The measure of an angle is 9 more than
2 times its complement. Find both angles.
ALL ANSWERS ARE POSTED ON-LINE AND IN THE CLASSROOM.
BE SURE TO CHECK THEM BEFORE CLASS STARTS.
4.1 & 4.2 Worksheet Answers
Make sure you showed ALL of your work!
M
1.
16.
17.
N
2.
T
G
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
18.
O
D
48
#1  50
# 2  40
#3  50
#1  79
# 2  56
#3  34
x  33 , Acute 
x  20 , Obtuse 
x  90 , Right 
x  39 , m  70
x  10 , m  109
x  37 , m  143
U , CPCTC
AC , CPCTC
8 , CPCTC
T  66 , CPCTC
19.
2 , //  AIA 
BD , Reflexive
CDB , Defn  
2 , VA    ' s
JKH , Defn  
AB , Defn Isos 
BC , Defn midpoint
AC , Reflexive
DCA , All rt s 
BAC , Defn  
Right s , Defn 
2 , All rt s 
4 , Defn  Bisector
WZ , Reflexive
WZX , Defn  
20.
21.
22.
4 10
3 13
 5,  2
24.
3
 5
 ,  
2
 2
1,  23
25.
 13, 15
26.
27.
35 , 145
27 , 63
23.
4.3 & 4.4 HOMEWORK
Triangle Proofs – Day 1
Name_______________________________#_______
Use the markings in each diagram to determine what triangle congruence pattern is shown.
SSS SAS
ASA
AAS
or
none
1.
2.
3.
4.
5.
6.
7.
8.
9.
Answer the following for each diagram. Be sure to READ each step completely.
10.
D
a.
G
A
O
C
MOP 
T
Is there enough information to prove the triangles
congruent?
Yes or No
*If there is not enough information to prove the
triangles are congruent, then circle:
congruence cannot be determined
b.
Circle the reason that proves the triangles are
SSS,
SAS,
AAS,
.
ASA
c. IF the triangles were congruent, complete the
statement:
DOG 
11.
A
M
E
O
P
a.
R
Is there enough information to prove the triangles
congruent?
Yes or No
*If there is not enough information to prove the
triangles are congruent, then circle:
congruence cannot be determined
b.
Circle the reason that proves the triangles are
SSS,
SAS,
AAS,
ASA
c. IF the triangles were congruent, complete the
statement:
.
12.
13.
N
P
A
T
M
a.
Is there enough information to prove the triangles
congruent?
Yes or No
a.
*If there is not enough information to prove the
triangles are congruent, then circle:
*If there is not enough information to prove the
triangles are congruent, then circle:
congruence cannot be determined
b.
Circle the reason that proves the triangles are
SSS,
SAS,
AAS,
ASA
c. IF the triangles were congruent, complete the
statement:
BIE 
Is there enough information to prove the triangles
congruent?
Yes or No
congruence cannot be determined
.
b.
Circle the reason that proves the triangles are
SSS,
SAS,
AAS,
ASA
c. IF the triangles were congruent, complete the
statement:
MAP 
.
Answer the following for each diagram. Be sure to READ each step completely.
Given:
14.
AB  ED
a.
Mark the diagram with the given information.
b.
List pairs of corresponding parts that you know are
congruent, including the given.
BC  DC
B  D
Be sure to mark any new information on your diagram.
D
B
S or A ____________________________
Why?_____________________
A
E
C
S or A ____________________________
Why?_____________________
S or A ____________________________
Why?_____________________
c.
What special pattern in marked in the triangles?
SSS,
d.
SAS,
AAS,
ASA, NONE (circle one)
If the triangles are congruent, complete the statement:
ABC 
15.
W  C
WA  CA
Given:
a.
Mark the diagram with the given information.
b.
List pairs of corresponding parts that you know are
congruent, including the given.
Be sure to mark any new information on your diagram.
M
S or A ____________________________
Why?_____________________
A
W
S or A ____________________________
C
Why?_____________________
S or A ____________________________
Why?_____________________
P
c.
What special pattern in marked in the triangles?
SSS,
d.
SAS,
AAS,
ASA, NONE (circle one)
If the triangles are congruent, complete the statement:
PAW 
16.
AC  DC ;
Given:
A   D
a.
Mark the diagram with the given information.
b.
List pairs of corresponding parts that you know are
congruent, including the given.
Be sure to mark any new information on your diagram.
S or A ____________________________
B
Why?_____________________
S or A ____________________________
Why?_____________________
S or A ____________________________
A
C
D
Why?_____________________
c.
What special pattern in marked in the triangles?
SSS,
d.
SAS,
AAS,
ASA, NONE (circle one)
If the triangles are congruent, complete the statement:
ABC 
17.
Given:
GE // HM
G  M
G
Prove: EGH  ______________
Statements
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
5.
EGH  _________
5.
E
H
M
HINT: This step should be a triangle congruence!!!
18.
Given: AS bisects MP ; 1  2
A
Prove: MAS  ____________
Statements
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
S or A
5.
5.
6.
18.
MAS  ________
Given:
M
1
2
P
S
HINT: This step should be a triangle congruence!!!
6.
A
M
A is the midpoint of MR
1
KA  RM
R
2
Prove: AKM  _________
Statements
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
S or A
5.
5.
S or A
6.
6.
7.
AKM  _________
7.
K
HINT: This step should be a triangle congruence!!!
4.3 & 4.4 HOMEWORK ANSWERS – Day , Introduction to Triangle Proofs
1.
2.
3.
4.
5.
6.
7.
8.
9.
SSS
SAS
ASA
None (no AAA)
AAS
None (no donkeys)
SAS
None (no AAA)
ASA
10. AAS,
17. (A) 1.
3.
(A) 4. GEH
5.
18. 1.
2.
TAC
14. (S)
AB  ED , Why?
(S)
BC  DC , Why?
Given
19. 1. A is the midpoint of
's 
MA  RA , Defn midpoint
2. KA  RM , Given
3. 1 & 2 Right  ' s ,
Defn  lines
(A) 4. 1   2 ,
All Right  ' s 
7.
16. TRIANGLES CANNOT BE PROVED CONGRUENT….
…YET!!, Here is what we will show next class 
A   D ,
Why? Given
AC  DC ,
Why? Given
** (S) AB  BD ,
Why? Base  ' s
AK  AK , Reflexive
AKM  AKR , SAS
(S) 5.
MAC

** c & d. Yes by SAS
DBC
MR , Given
(S) 1.
c & d. Yes by ASA
**e.
AS  AS , Reflexive
MAS  PAS , SAS
6.
W  C , Why? Given
(S) WA  CA , Why? Given
(A) WAP  CAM
(S)
bisect
Defn Segment Bisector
(S) 5.
15. (A)
(A)
MP , Given
S is midpoint of MP .
AS
Given
EDC
e.
 MHE , //  AIA 
EGH  HME , AAS
(S) 3. MS  PS ,
Defn Midpoint
(A) 4. 1   2 , Given
(A) B  D , Why? Given
c & d. Yes by SAS
Why? Vert
HE  HE , Reflexive
GE // HM , Given
(S) 2.
11. SSS, AER
12. SAS, KEI
13. Cannot determine (no AAA)
e.
G  M , Given
Sides
Geometry Homework
Section 4.3 & 4.4 – Triangle Proofs (Day 2)
Name_________________________________#_________
Reason Bank
Given
//  CIA supp.
Defn 
Linear Pair Postulate
Defn Supp s
Defn Seg. Bisector
Segment Addition
Defn Midpt
Defn Isosceles 
Reflexive
Base s    sides
//  AEA 
//  AIA 
//  CA 
Reflexive
 ' s Supp to same ->   ' s
 ' s Comp to same ->   ' s
Vertical s 
Defn  Bisector
All right s 
2 s in one   to 2
s in one    3rd s 
Isos   base s 
Add/Sub property
Substitution
Defn 
CPCTC
AAS
ASA
SSS
SAS
HL
____
1.
Given: ΔCDE is an isosceles Δ ; G is the mdpt of CE
D
Prove: CDG  EDG
C
Statements
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
S or A
5.
5.
6.
7.
6.
CDG  EDG
7.
G
E
HINT: This step should be a triangle congruence!!!
____
2.
____
____
____
Given: GH // KJ ; GK // HJ
G
1
Prove: ΔGJK  ΔJGH
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
S or A
5.
5.
ΔGJK  ΔJGH
____
3.
____ ____
4
3
K
Statements
6.
H
2
J
6.
____
Given: DE // JK , DK bisects JE
J
2
Prove: ΔEGD  ΔJGK
D
1
3
G
4
Statements
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
S or A
5.
5.
S or A
6.
6.
7.
ΔEGD  ΔJGK
7.
E
K
S
____
4.
____
____
Given: TS  SF  FH ;
TSF and HTS are right s
____
F
T
____
Prove: HS  TF
H
Hint: Cover up FH to make the picture “easier.”
Statements
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
5. TSF  HTF
____
6.
5.
HINT: This step should be a triangle congruence!!!
5.
____
HS  TF
6.
Given: O is the midpoint of AY
O is the midpoint of ZX
X
Y
Prove: Y  A
5
O
6
Statements
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
S or A
5.
5.
5.
6.
Z
A
HINT: This step should be a triangle congruence!!!
6.
7.
Complete the following proofs 6.
no hints!
Do your best!
G
Given: FH bisects GFJ ;
FH bisects GHJ ;
F
H
Prove: FG  FJ
J
Statements
7.
Reasons
D
Given: AD  CD
B is the midpoint of AC ;
Prove: ADB  CDB
A
Statements
Reasons
B
C
Geometry Homework
Section 4.3 & 4.4 – Triangle Proofs (Day 3)
____
1.
____
____
Name_________________________________#_________
____
Given: AC  GC, EC bi sec ts AG,
A
Prove: ΔGEC  ΔAEC
C
E
Statements
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
S or A
5.
5.
6.
ΔGEC  ΔAEC
G
6.
____
2.
Given: QS bisects RST; R  T
R
Prove: ΔQRS  ΔQTS
Q
S
T
Statements
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
5.
ΔQRS  ΔQTS
5.
3.
____
____
____
H
Given: EF  HF , G is the midpoint of EH
Prove: ΔEFG  ΔHFG
E
Statements
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
ΔEFG  ΔHFG
5.
____
4.
G
F
5.
____
Given: V  S; TV  QS
____
S
____
Prove: VR  SR
R
T
Statements
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
4.
4.
Q
V
____
____
5. VR  SR
5.
HINT: This step should be a triangle congruence!!!
Try some overlapping triangle proofs!
5.
Given: N  S; OUB is isosceles
Prove: BON  OBS
O
1
U
2
Statements
N and _________
ON and _________
1 and _________
BN and _________
BON and _________
You need 3 to prove the  's 
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
6.
BO and _________
S
B
5.
The corresponding parts are…
N
BON  OBS
5.
HINT: This step should be a triangle congruence!!!
Given: 1  2, IHP  IPH
Prove: HIL  PIE
I
E
1
H
The corresponding parts are…
2 L
IH and _________
1 and _________
LH and _________
IHL and _________
You need 3 to prove the  's 
Reasons
S or A
1.
1.
S or A
2.
2.
S or A
3.
3.
S or A
4.
4.
5.
I and _________
P
Statements
congruence!!!
IL and _________
HIL  PIE
5.
HINT: This step should be a triangle
Geometry HW – Section 4.6
Isosceles, Equilateral, and Right Triangles
Name________________________#_______
Decide whether enough information is given to prove that the triangles are congruent. Explain your
answer.
R
1.
D
A
2.
Q
3.
R
S
T
B
S
U
C
U
T
Use the statements and reasons from the boxes at the right. Note that although there may be alternate
approaches to these proofs you must use the options presented. There may be extra statements and/or
reasons for some proofs.
4.
Given:
1  2
____
T
1
____
TP  RA
Prove:
R
4 2
3
Possible
Statements
M
3  4
P
HINT: Look at the statements to find the
triangles you need to prove congruent.
Outline those triangles.
Statement
Reason
A
____
____
____
____
TR  TR
1  2
TMP  RMA
P  A
TMP  RMA
TP  RA
3  4
____
____
TM  RM
Possible Reasons
AAS
CPCTC
Vertical s 
Given
ASA
Given
Base ’s    Sides
Isosceles  Base ’s 
5.
Given:
____
A
____
AB  AC
____
M is mdpt of BC
Prove:
____
AM bisects  BAC
B
Statement
C
M
Reason
Statements
Reasons
____
Reflexive
Given
Defn  bisector
ASA
SSS
Given
Defn seg. Bisector
CPCTC
Defn midpoint
AM bisects BAC
BAM  CAM
____
____
____
____
____
____
AM  AM
BM  CM
ABM  ACM
AB  AC
B  C
____
M is mdpt of BC
BMA  CMA
Solve for the variable(s) in each picture. Show all your work! PICTURES ARE NOT DRAWN TO
SCALE.
6.
 3x  6  
 2x  11 
7.
8.
3y  
 3x  
 y  7 
 x  2 
 4x  10 
2x
9.
10.
11.
54
y
x
x
y
x
72
Decide whether enough information is given to prove that the triangles are congruent. Explain your
answer.
12.
13.
P
T
14.
O
J
F
R
E
A
M
K
L
M
N
C
D
N
U
15.
B
16.
A
17.
T
O
R
M
O
S
P
B
C
D
Use the direction for questions 4 & 5 to complete the proofs below. Remember , there may be extra
statements and/or reasons for some proofs.
18.
Given:
____
____
BD  BE
Prove:
B
 BAC   BCA
BDC  BEA
HINT: Look at the statements to find the
triangles you need to prove congruent.
Outline those triangles.
D
A
Statement
E
C
Reason
Possible
Statements
____
____
____
____
____
____
BD  BE
D  E
DC  EA
BDC  BEA
BAC  BCA
BA  BC
B  B
BAE  BCD
Possible Reasons
AAS
Given
Given
SAS
CPCTC
Reflexive
Isosceles   Base ’s 
Base ’s    Sides
19.
Given:
____
A
BD bisects  ADC
____
____
DB  AC
Prove:
B 3
4
ADC is isosceles
1
2
D
C
Statement
Reason
Possible
Statements
ADC is isosceles
ADB  CDB
3 & 4 are rt ’s
1  2
3  4
____
____
____
____
AD  CD
DB  DB
____
BD bisects  ADC
____
Possible
Reasons
Defn of 
Reflexive
Defn  bisector
Defn of isos 
Given
ASA
All rt ’s 
CPCTC
Given
____
DB  AC
Solve for the missing variables in each diagram. Remember to use the properties above.
Label each angle once you get it figured out!!! Only work with one triangle at time.
Pictures are not drawn to scale.
20.
21.
y
 3x  8  
x
 2x  20 
22.
5x 10 
23.
 4x  10 
40
 4y  
3x  10 
24.
25.
3x  13 
 4x 10 
 7 x  3 
 2x  12 
4 y
 2 y  3 
26.
27.
 x  2 
50
(2 y )
 2x  11 
 3x  
y
142
Mixed Review – (Sections 1.3 – 1.5)
____
____
28.
Given U(12, -1) is the midpoint of HG and
H(-4, 8). Find the coordinates of the other
endpoint.
29.
B is the midpoint of UG .
BU = (7x + 6), GU = (2x +36).
Find BU.
30.
T is in the interior of BIS . Find x.
31.
Find AB. A(-7, 7) B(-4, -11)
mBIT  (4x  3), mSIT  (3x  8)
mBIS  68
32.
UP bisects TUG , mTUP  (13x  39)
and mPUG  52 . Solve for x
34.
S is the midpoint of AH . Find x.
SH = 37 and AH = (14x - 24).
33.
J is between C and R. Find RJ.
CJ = (7 - 3x), JR = (5x -2),
CR = (7x - 15)
35.
Find the midpoint of AT if
A( 16, -2) and T(-2, 4).
____
____
4.6 Isosceles, Equilateral, and Right Triangles HW Answers
1.
2.
3.
4.
5.
yes, HL or AAS
yes, AAS
no, NEI
1  2, Given
TP  RA , Given
 TMP   RMA, VA  
TMP  RMA. AAS
TM  RM , CPCTC
3  4, Isosceles 
Base ’s 
AB  AC , Given
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
M is mdpt of BC , Given
BM  CM , Defn midpoint
AM  AM , Reflexive
ABM  ACM, SSS
BAM  CAM, CPCTC
AM bisects BAC, Defn 
bisector
6.
7.
8.
x = 25
x = 15, y = 38
x = 29, y = 51
19.
x = 45, y = 135
x = 72, y = 36
x = 31.5
yes, HL
yes, AAS
yes, no (Look closely at the
markings.)
yes, AAS or ASA
no, NEI
yes, SAS
BAC  BCA, Given
BD  BE , Given
B  B, Reflexive
BAE  BCD, AAS
BDC  BEA, CPCTC
BD bisects ADC, Given
1  2, Defn  bisector
DB  AC , Given
3 & 4 are rt ’s, Def 
3  4, All rt ’s 
DB  DB , Reflexive
ADB  CDB, ASA
AD  CD , CPCTC
ADC is isos, Defn of isos 
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
x = 60, y = 120
x = 18
x = 22.5
x = 10, y = 20
x = 4, y = 11
x = 11, y = 11
x = 15, y = 10
x = 32, y = 19
 28, 10
x  2, BU  20
x 9
3 37
x7
x  4, RJ  18
x7
 7, 1
Geometry Homework
Chapter 4 Review
Name____________________________________#_______
YOU MUST SHOW ALL YOUR WORK TO RECEIVE CREDIT FOR THIS PACKET.
(If you use a calculator, write it down!!!)
Determine if the statement is (S)ometimes, (A)lways, or (N)ever true
1.
________A scalene triangle is isosceles.
6.
________An isosceles triangle is acute.
2.
________An isosceles triangle is equilateral.
7.
________A right triangle is equilateral.
3.
________A scalene triangle is right.
8.
________A right triangle is isosceles.
4.
________An equilateral triangle is acute.
9.
________An acute triangle is equilateral.
5.
________An isosceles triangle is obtuse.
10.
________A scalene triangle is obtuse.
Given the picture at the right, determine the measure of each angle. (Hint: Label the entire picture first.)
11.
m2 ________
12.
m1 ________
13.
m9 ________
D
2 1
35
C
120
3
4
14.
m7 ________
15.
m4 ________
16.
m8 ________
17.
Given: ABC is isosceles with vertex C. If mA   3x  6  and mC   2 x  . Solve for x and
55
A
7
9
8
B
find mB . (Hint: Draw and label a picture.)
18.
Given an isosceles triangle with vertex angle that measures 70, what is the measure of a base angle?
19.
Given: ABC  XYZ, AB = 38, YZ = 28, and XY = 5x + 8. Solve for x . (Hint: What parts are
corresponding?)
ABC is an isosceles triangle with vertex angle B. AB = 5x – 28, AC = x+ 5, and BC = 2x + 11. Solve
for x and find the length of the base. (Hint: Draw and label a picture.)
20.
Determine if the given triangles are congruent. If possible, state the theorem or postulate to justify your
answer.
21.
________
22.
________
23.
________
24.
________
25.
________
26.
________
27.
________
28.
________
31.
________
Solve for x for each picture.
29.
________
30.
________
◦
(x+5)
◦
(3x - 10)
32.
________
x  11
 2 x  5 
◦
(115)
2x
5x  3
33.
2x  8
 3x 1 
________
34.
________
5x  
7 x 12
3x  8
 7 x  9 
 27  
35.
________
36.
________
37.
________
 2 x  2 
15
 4x  
40
5x  
 3x  8  
135
34
38.
________
39.
________
 x  3 
 2x  16 
 x  15 
 2 x  56 
 x  28 
 2x  1 
Use the picture at the right to answer the following questions.
40.
Name the all the right triangles.
B
41.
Name all the isosceles triangles.
42.
Which triangle(s) is obtuse?
43.
Which angle(s) is opposite BE?
44.
Which side(s) is opposite DCA
D
A
E
F
C
Complete the following proofs.
Given:
45.
P
QP // ST
R is the mdpt of QT
Prove:
R
Q
PR  SR
T
S
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
A
Given:
46.
DA bisects BAC
1 2
AD  BC
Prove: B  C
B
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
3 4
D
C
Given: BE // AD
BE  AD
AE bisects BC
47.
B
A
1
Prove: ABE  CAD
2
E
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
Bonus: Use the proof above to answer this “What if…?”
Prove:
AE // CD
Statements
Reasons
8.
8.
9.
9.
C
3
4
D
Ch 4 Review Pkt Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
N
S
S
A
S
S
N
S
S
S
65
25
120
35
60
25
x=24, mB=66
55
6
x=13, base=18
NEI
ASA
SAS
ASA OR AAS
NEI
ASA
SSS
HL
30
6
1
19
5
12
14
21
19
25
15
AFC, DFE, AFD,
CFE
AFC, ABC
NONE
BAE
DA
As you check the proofs, remember
that different orders are possible, but
certain steps must stay together. I
have inserted blank lines so you can
see the steps that should be grouped.
I have also inserted the “line” to show
what steps must come BEFORE you
state that the triangles are congruent.
45.
Alternate methods are
possible for this proof. Please
have Mrs. G look at yours if
you are unsure. 
1. QP // ST , Given
2. Q  T , //  AIA 
3. R midpoint QT ,
Given
4. QR  TR , Defn mdpt
5. QRP  TRS , VA 
6. QRP  TRS , ASA
7. PR  SR , CPCTC
46.
1. DA bisects BAC ,
Given
2. 1  2 ,
Defn  bisector
3. AD  BC , Given
4. 3& 4 Rt , Defn 
5. 3  4 , All Rt ’s 
6. AD  AD , Reflexive
7. ABD  ACD , ASA
8. B  C , CPCTC
47.
1. BE // AD , Given
2. 1  3 , //  AIA 
3. BE  AD , Given
4. AE bisects BC , Given
5. A mdpt BC ,
Defn Seg Bisector
6. BA  AC , Defn mdpt
7. BEA  ADC , SAS
8. 2  4 , CPCTC
9. AE // CD , AIA   //