Math 9
Similar Triangles
MOM Page 213-215
Lesson 3-7
Name SOLUTIONS
4. For each pair of similar triangles, list the
corresponding sides and angles.
Corresponding Sides and angles:
AB and ED
AandE
AC and EF
CandF
BC and DF
Band D
Corresponding Sides and angles:
PQ and ZX
PandZ
QR and XY
QandX
RR and ZY
Rand Y
5. Each pair of triangles are similar. In each case, state the measures of the angles that are not
marked.
A = 34
Q = 90
P = 56
D = 70
J = 70
G = 70
F = 40
K=M=S=T=U
60
6. Explain why each pair of triangles are similar. Find the values of x and y.
S  B
T  C
ThereforeR  A
J  V
Q  R
ThereforeC  C
Therefore RST is similar to ABC
Therefore JCQ is similar to VAR
RT
RS
ST


AC AB BC
3 y
x


5 13 12
3 y

5 13
3(13)  5 y
39  5 y
JC JQ CQ


VA VR AR
4 8 5


x 12 y
4 8

x 12
4(12)  8 x
48  8 x
7. 8  y
3 x

5 12
3(12)  5 x
36  5 x
7 .2  x
8 5

12 y
5(12)  8 y
60  8 y
7 .5  y
6x
8. State the ratios of the corresponding sides of each pair of similar triangles. Find each value of
x.
xy yz xz


pq qr pr
11 yz x


6 qr 4
11 x

6 4
4(11)  6 x
44  6 x
7 .3  x
TN NC TC


EQ QS ES
7 NC 10


12 QS
x
7 10

12 x
7 x  12(10)
7 x  120
x  17.14
9. State which triangles are similar. Find the values of x and y.
XYZ  PQR
XY YZ
XZ


PQ QR PR
x 6 4


8 10 y
10 x  8(6)
XY YZ
XZ


PQ QR PR
x 6 4


8 10 y
6 y  4(10)
10 x  48
x  4 .8
6 y  40
y  6.67
PST  PQR
PS PT ST


PQ PR QR
10  y x  3 24


10
x
20
10  y (20)  10(24)
200  20 y  240
20 y  40
y2
10. For each diagram, name two similar triangles.
Find the length of CE.
CAB  CED
ACE  ABD
CA CB


CE CD
10 CB


CE CD
10 8

x 4
8 x  10(4)
8 x  40
x5
AC AE CE


AB AD BD
AC 21 CE


BC
7
4
21 x

7 4
7 x  21( 4)
7 x  84
x  12
AB
ED
8
4
PS PT ST


PQ PR QR
10  y x  3 24


10
x
20
20( x  3)  24( x)
20 x  60  24 x
60  4 x
15  x
11. For this picture of an ironing board, explain how you know that AEC is similar to BED .
The floor and the board are parallel lines
Therefore:
 A =  B ( Z pattern)
 C =  D ( Z pattern)
 E =  E (Opposite angles)
Therefore:
AEC  BED
12. Explain why each pair of triangles are similar. Find the values of x and y.
F  B
E  C
ThereforeD  A
K  G
L  H
ThereforeJ  I
Therefore FED is similar to BCA
Therefore KLJ is similar to GHI
FE FD DE


BC BA AC
6
x
8


y 15 12
6 8

y 12
6(12)  8 y
72  8 y
9 y
KL KJ JL


GH GI IH
18 x 15
 
12 6 y
18 15

12 y
12(15)  18 y
180  18 y
10  y
x
8

15 12
8(15)  12 x
120  12 x
10  x
x 18

6 12
6(18)  12 x
108  12 x
9x
13. Name a pair of similar triangles in each diagram. Explain how you know the triangles are
similar.
P  T
R  S
Q  Q
Alternate Interior
Alternate Interior
Opposite Angles
X  V
Y  W
U  U
Corresponding Angles
Corresponding Angles
Shared Angle
XYU  VWU
PRQ  TSQ
15. For each diagram, name two similar triangles. Find the values of x and y.
RSQ is similar to RTP
BCD is similar to FED
BC CD BD


FE ED FD
y 1.2 1.4


4.8
x
5.6
y
1.4

4.8 5.6
1.4(4.8)  5.6( y )
6.72  5.6 y
1.2  y
1. 2 1. 4

x
5.6
1.2(5.6)  1.4 x
6.72  1.4 x
4.8  x
RS SQ RQ


RT TP RP
12
x 6


y  12 15 9
12
6

y  12 9
6( y  12)  12(9)
6 y  72  108
x 6

15 9
6(15)  9 x
90  9 x
10  x
6 y  108  72
6 y  180
y  30
GHJ is similar to LKJ
GH GJ HJ


LK
LJ KJ
6.10 16.2
y


x
12.96 12
16.2
y

12.96 12
16.2(12)  12.96( y )
194.4  12.96 y
15  y
6.10 16.2

x
12.96
16.2 x  6.1(12.96)
16.2 x  79.056
16. In each diagram, the triangles are similar. For each pair of triangles, write the ratio of sides
AB
that is equal to
.
BC
FD
DE
DE
EF
ED
DF
17. The two triangles in each diagram are similar. Find each
length represented by x.
21 x

42 14
42( x)  21(14)
42 x  294
x7
2 .0 1 .5

2 .4
x
1.5( 2.4)  2( x )
3 .6  2 x
x  1 .8
x 10

15 20
20( x)  10(15)
20 x  150
x  7.5
19. Name a pair of similar triangles in this figure. Explain how you know the triangles are
similar.
DGE is similar to FDE
Because: Both have a right angle, and  E is common to both triangles.
DGF is similar to EDF
Because: Both have a right angle, and  E is common to both triangles
DGE is similar to FGD
Because: Both have a right angle, and  P is common to both triangles
20. In each figure, name a pair of similar triangles. Explain how you know they are similar.
Find the values of x and y.
ADC is similar to ABE
Because: A=A,  D=B, C=E
IFG is similar to IHF
20 18 8  y


8
x
6
20 18

8
x
8(18)  20 x
15
y 13.2  y
 
13.2 x
15
15 13.2  y

13.2
15
15(15)  13.2(13.2  y )
144  20 x
x  7 .2
20 8  y

8
6
8(8  y )  20(6)
64  8 y  120
8 y  120  64
8 y  56
y7
BDE is similar to BAC
15
8
17


17  y 12 15  x
8
15

12 17  y
12(15)  8(17  y )
180  136  8 y
180  136  8 y
44  8 y
5.5  y
8
17

12 15  x
12(17)  8(15  x)
204  120  8 x
204  120  8 x
84  8 x
10.5  x
3.84 15

x
13.2
3.84(13.2)  15( x)
50.76  15 x
3.38  x
225  174.24  13.2 y
225  174.24  13.2 y
50.76  13.2 y
3.84  y
RQS is similar to RPQ
4
y x
 
10 8 4
4
y

10 8
4(8)  10( y )
32  10 y
3 .2  y
4 x

10 4
4( 4)  10 x
16  10 x
1 .6  x
MOM Page 221-223
Name SOLUTIONS
4. State which triangles are similar? Find the values of
x and y.
 Y =  U given
 w =  S given
 X =  T Sum of Triangle
 D =  Z given
 E =  A given
 C =  B Sum of Tri.
 F =  K given
 G = I given
 H =  J Sum of Tri.
YWX  UST
DEC  ZAB
FGH  KIJ
YW WX YX


US
ST UT
y 6 x
 
10 8 12
DE EC DC


ZA AB ZB
18 9 20


y 12
x
FG GH FH


KI
IJ
KJ
x 8 12
 
18 y 15
y 6

10 8
10(6)  8 y
60  8 y
7.5  y
6 x

8 12
8 x  6(12)
8 x  72
x9
18 9

y 12
9 y  18(12)
9 y  216
y  24
9 20

12
x
9 x  12( 20)
9 x  240
x  26.7
x 12

18 15
15 x  18(12)
15 x  216
x  14.4
12 8

15 y
12 y  15(8)
12 y  120
y  10
5. In each figure, state which triangles are similar, then find the value of x.
 A =  D Corresponding angles
 B= E Corresponding angles
 C =  C Common Angle.
 L =  N Alternate Interior
 K =  P Alternater Interior
 M =  M Opposite angles
ABC  DEC
LKM  NPM
AB BC AC


DE EC DC
x 12 AC


6 8 DC
LK KM LM


NP PM NM
x 12 LM


5 4 NM
x 12

6 8
8 x  6(12)
8 x  72
x9
x 12

5 4
4 x  5(12)
4 x  60
x  15
 V =  Z Alternate Interior
 W = Y Alternate Interior
 X =  X Opposite angles
 E =  D Corresponding angles
 B= C
Corresponding angles
 A =  A Common Angle.
VWX  ZYX
EBA  DCA
VW WX VX


ZY
YX
ZX
6 WX
x


15 YX
25
EB BA EA


DC CA DA
9
x
EA


18 6  x DA
6
x

15 25
15 x  6( 25)
15 x  150
x  10
9
x

18 6  x
9(6  x)  18 x
54  9 x  18 x
54  9 x
6x
6. To find the distance PQ across a farm pond, Marty marks out points R and S so that RS is
parallel to PQ. By measuring, she finds that RS = 5.7 m, OP = 19.5 m, and OS = 4.2 m.
What is the distance PQ?
 P =  S,  Q =  R,  O =  O
Therefore: PQO  SRO
PQ QO PO


x
19.5
SR RO SO

5 .7 4 .2
x
QO 19.5


4.2 x  (5.7)(19.5)
5.7 RO
4 .2
4.2 x  111.15
x  26.46
The distance of PQ is 26.46 m.
7. Two trees cast shadows as shown. How tall is the evergreen tree?
14 5

x 8
5 x  14(8)
5 x  112
x  22.4
The evergreen tree is 22.4 m tall.
PAGE 222
10. The shadow of a telephone relay tower is 32.0 m long on level ground. At the same time, a
boy 1.8 m tall casts a shadow 1.5 m long. What is the height of the tower?
1.8 1.5

h
32
1.5h  1.8(32)
1.5h  57.6
h  38.4
The height of the tower is 38.4 m.
11. Karen is 37.5 m from a church. She finds that a pencil, 4.8 cm long, which is held with its
base 60 mm from her eye, just blocks the church from her sight. How high is the church?
37.5
h

0.06 0.48
0.06h  37.5(0.48)
0.06h  18
h  300
The height of the church is 300 m.
12. In each figure, state which triangles are similar? Find the values of x and y.
 A =  B corresponding angles
 E =  D corresponding angles
 C =  C common angle
x 12 y  10


6 8
10
AEC  BDC
x 12

6 8
8 x  72
12 y  10

8
10
8 y  80  120
x9
8 y  40
y5
14. To find the distance across a river, Elinka uses the sketch and the measurements shown
below. Find the distance across the river.
 A =  A common angle
 D =  B corresponding
 E =  C corresponding
Therefore
ADE  ABC
AD DE AE


AB BC AC
x
24 AE


x  48 60 AC
x
24

x  48 60
60 x  24( x  48)
60 x  24 x  1152
36 x  1152
x  32
The distance across
the river is 32 m.
15. To calculate the height of a building, Rudolf uses the height of a pole and the length of the
shadows cast by the pole and the building, as shown in the diagram. How tall is the
building?
3 . 5 1 .5

x
6
1.5 x  3.5(6)
The building is
14 m tall.
1.5 x  21
x  14
PAGE 223
17. a pole 3.8 m high casts a shadow that measures 1.3 m. A nearby tree casts a shadow 7.8 m
long. Find the height of the tree.
3 .8 1 .3

x
7.8
1.3 x  3.8(7.8)
1.3 x  29.64
x  22.8
The height of the
tree is 22.8 m.
19. To determine the height of a tree, Jerry places a 2 m rod 24 m from the tree. He finds that he
can align the top of the rod with top of the tree when he stands 1.9 m from the rod. Jerry’s
eyes are 1.6 m from the ground. What is the height of the tree?
0.4 1.9

x
24
1.9 x  0.4(25.9)
1.9 x  10.36
x  5.45
x  2  7.45
The height of the tree
is 7.45m.