Module 2 similarity

Module 17
Similar Triangles
What this module is about?
This module is about similar triangles, definition and similarity
theorems. As you go over the exercises you will develop your skills
in determining if two triangles are similar and finding the length of a
side or measure of an angle of a triangle.
What are you expected to learn?
This module is designed for you to:
1. apply the definition of similar triangles in:
a. determining if two triangles are similar
b. finding the length of a side or measure of an angle of a
triangle
2. verify the Similarity Theorems:
a. AAA Similarity
b. SAS Similarity
c. SSS Similarity
3. apply the properties of similar triangles and the
proportionality theorems to calculate lengths of certain line
segments.
How much do you know?
A. For each pair of triangles, indicate whether the two triangles are
similar or not. If they are similar, state the similarity theorem or
definition that supports your answer.
1.
2
450
3
4
450
6

2.
3.
4.
5.
B.
6 – 10. The length of the sides of a triangle are 14, 8 and 6. Find the
length of the two sides of a similar triangle if the length of the shortest
side is 12.
3
3
1
2
3
3
2
2
800
500
500
6 4
3 2
4
12

Lesson 1
Definition of Similar Triangles
Similar Triangles
Two triangles are similar if corresponding angles are congruent
and corresponding sides are proportional.
Examples:
1. If in ∆ ABC and ∆ XYZ
∠ A≅ ∠ X
∠ B≅ ∠ Y
∠ C≅ ∠ Z
XY
AB
=
YZ
BC
=
XZ
AC
then ∆ABC ∼ ∆XYZ (Triangle ABC is similar to triangle XYZ)
2. ∆BMP ∼ ∆SEC
Find the value of x and y
Since the corresponding sides are proportional
SE
BM
=
EC
MP
=
SC
BP
x
3
=
8
4
=
y
5
x
3
=
2
1
=
y
5
x
3
=
2
1
;
2
1
=
y
5
x = 6 y = 10
A
B
C
X
Y
Z
B P
M
3 4
5
S
E
C
x
y
8

Exercises:
A. Given: MOL∆ ∼ REY∆
Fill the blanks.
1) M∠ ≅ ______
2) O∠ ≅ ______
3) _____ ≅ Y∠
4)
RY
ML
=
?
MO
5)
RY
ML
=
?
LO
6)
3
2
=
15
?
7)
6
4
=
?
2
B. State whether the proportion is correct for the indicated similar
triangles
1. RST∆ ∼ XYZ∆ 4. DEF∆ ~ HIS∆
XY
RS
=
YZ
ST
HI
DE
=
IJ
EF
2. ABC∆ ~ DEF∆ 5. KLM∆ ~ PQR∆
DE
AB
=
EF
BC
PR
KM
=
QR
LM
3. RST∆ ~ LMK∆ 6. XYZ∆ ~ UVW∆
LM
RT
=
MK
ST
UV
XY
=
UW
XZ
L O
M
2
4
5
Y
R
E
3
6
2
15

Complete the proportions
7. ABC∆ ~ DEF∆ 9. RST∆ ~ XYZ∆
?
AB
=
?
BC
=
?
AC
?
XY
=
?
XZ
=
?
YZ
8. KLM∆ ~ RST∆ 10. MNO∆ ~ VWX∆
KL
?
=
LM
?
=
KM
?
?
VX
=
?
VW
=
?
WX
C. Find the missing length
1. ABC∆ ~ XYZ∆
find b and a
2. SIM∆ ~ PON∆
SI = 6cm. IM = 4cm. SM= 8cm,
Find the lengths of the sides of PON∆ if the ratio of the lengths of
the corresponding sides is 1:3.
A
B
C
3 a
b
X Z
Y
5 10
15
S
6cm.
I
4cm.
M
8cm
P
O
N

Lesson 2
AAA Similarity Theorem
If in two triangles the corresponding angles are congruent, then
the two triangles are similar.
If BOS∆ ↔ VIC∆
B∠ ≅ V∠ , O∠ ≅ I∠ , S∠ ≅ C∠
then BOS∆ ~ VIC∆
AA Similarity
If two angles of one triangle are congruent to the corresponding
two angles of another triangle, the triangles are similar.
Given: A∠ ≅ O∠
J∠ ≅ B∠
Then: JAM∆ ~ BON∆
Examples:
Are the triangles similar by AA Similarity?
1.
B
O
S V
I
C
J
A
M
O
B N
330
470
1000
330
470
1000

2.
3.
4.
5.
410
390
600
300
950
950
600
600
600
600

SAS Similarity Theorem
If in two triangles two pairs of corresponding sides are
proportional and the included angles are congruent, then the triangles
are similar.
If ICE∆ ↔ BOX∆
OB
CI
=
OX
CE
and C∠ ≅ O∠
then ICE∆ ~ BOX∆
Examples:
AOB∆ ~ DOC∆ by SAS
since
DO
AO
=
CO
BO
and
AOB∠ ≅ DOC∠
Can you explain why is
RAT∆ ~ RAM∆ by SAS ?
I
C
E
B
O
X
B
A
O
D
C
9
6 12
8
R
A
T
M
33
4

SSS Similarity Theorem
If the two triangles three corresponding sides are proportional,
then the triangles are similar.
If SUN∆ ↔ BLK∆
BL
SU
=
LK
UN
=
BK
SN
then SUN∆ ~ BLK∆
Example:
1.
OB
SN
=
PE
NA
=
DE
SA
6
4
=
9
6
=
12
8
since the corresponding
sides are proportional then
NSAPOE ∆∆ ~ by SSS
Similarity.
2. Explain: Any two congruent triangles are similar.
S N
U
B K
L
P E
O
6 12
9
N A
S
84
6

Exercises:
A. Tell whether the two triangles are similar. Cite the Similarity
Postulate or Theorem to justify your answer. (Identical marks
indicate ≅ parts.)
1.
2.
3.
4.
6 8
10
3
5
4
6
8
3
4
3
6
6
3
9
6 8
10
12
15

5.
6.
7.
8.
6 6
66
5
5
3
3
2
8
84

9.
PS
QS
=
ST
RS
10.
B. Determine whether each pair of the following triangles are
similar by SAS, AAA, SSS or not at all.
1.
Q
R
S
P
T
750
1214
750
67

2.
3.
4.
5.
700
700
850
26
25
85012.5
13
60 60
9
16
6
14
21
24

C. Answer the following:
1. Two isosceles triangles have an angle of 500
. Does it follows that
the triangles are similar?
2. Two angles of BEL∆ have measures of 20 and 50. Two angles of
JAY∆ have measures of 30 and 100. IS JAYBEL ∆∆ ~ ?
3. Is it possible for two triangles to be similar if two angles of one
have measures 50 and 75, where as two angles of the other have
measures 55 and 70?
4. Two angles of have measures 40 and 80, where as the two angles
of the other have measures 60 and 80, are the two triangles
similar?
5 – 6. The lengths of the sides of a triangle are 12 and 15. If the
length of the shortest side of a similar triangle is 12, find the
lengths of the other two sides.
7- 8. In the figure, if AE = 8
AB=4, BC=10, ED=3
Find BD and DC.
9 – 10. Explain: Any two equilateral triangle are similar.
500
500
A
B
D
C
E

Lesson 3
Application of the Properties of Similar Triangles to Calculate
Lengths of certain line segments.
Examples:
DEFABC ∆∆ ~ . Find the missing measure.
1. AB = 36, BC = 24, DE = 48 Bm∠ = 110, Em∠ = 110; EF =_______.
DE
AB
=
EF
BC
48
36
=
x
24
36x = 1152
x = 32
EF = 32
2. AB = 38, BC =24, AC=30, DE=12, EF=16, DF= _______.
DE
AB
=
DF
AC
12
18
=
x
30
2
3
=
x
30
3x = 60
x = 20
DF = 20
3. AC = 15, DE=12, DF=20, =∠Am 35=∠Dm , AB =__________
DE
AB
=
DF
AC
12
x
=
20
15
12
x
=
4
3
4x = 36
x = 9
AB = 9

4. The lengths of the sides of a triangle are 14, 8 and 6. Find the
perimeter of a similar triangle if the length of its shortest side is 12.
Let x = the length of one side of a triangle
x = the length of the longest side of the triangle
12
6
=
x
8
2
1
=
x
8
x = 16
12
6
=
y
14
2
1
=
y
14
y = 28
P = 12 + 16 + 28
P = 56
5. On a level ground, a 5ft. person and a flagpole cast shadows of 10
feet and 60 feet respectively. What is the height of the flagpole?
Let x = the height of the flagpole
x
5
=
60
10
x
5
=
6
1
x = 30 feet the height of the flagpole

Exercises
1. A yardstick casts a shadow of 24in. at the same time an electric
post cast a shadow of 20ft. 8 in. What is the height of the electric
post?
2. Two triangles are similar. The lengths of the sides of one triangle
are 5, 12 and 13. Find the lengths of the missing sides of the
other triangle if its longest side is 39.
3. The perimeter of a triangle is 32cm. and the ratio of the sides is
3:6:7. Find the length of each side of the triangle.
4. A tall building at Makati casts a shadow of 12m. at the same time
a 7m. light pole cast a shadow of 3m. Find the height of the
building.
5. If the shadow of the tree is 20m. long and the shadow of the
person, who is 190cm. tall, is 250cm. long. How tall is the tree?
Let’s Summarize
1. Two triangles are similar if their vertices can be paired so that
corresponding angles are congruent and the lengths of
corresponding sides are proportional.
2. The AAA Similarity. If the corresponding of two triangles are
congruent, then the two triangles are similar.
3. The AA Similarity. If two pairs of corresponding angles of two
triangles are congruent, then the two triangles are similar.
4. The SSS Similarity. If the lengths of corresponding sides of two
triangles are proportional, then the two triangles are similar.
5. The SAS Similarity. If one pair of corresponding angles of two
triangles are congruent and the lengths of the corresponding sides
that include these angles are proportional, then the two triangles
are similar.

What have you learned?
A. Given the figures and the information below can you
conclude that RPSKMN ∆∆ ~ ?
If so what Similarity Theorem?
1. 80=∠Mm 80=∠Pm
2. 80=∠Mm , 100=∠+∠ SmRm
3. RS = 26
KN = 40
B. Find the missing side
4.
5.
6 – 7. If the shadow of the tree is 14cm. long and the shadow of the
person who is 1.8m. tall is 4m long, how tall is the tree?
8 – 10. A pole 3m. high has a shadow 5m long when the shadow of a
nearby building is 110m. long. How tall is the building?
K
24
M
36
N
R
16
P
24
S
24
x 5
6
16
20
3 4
5
9 12
X =?

Key
How much do you know?
1. yes, SAS
2. No
3. yes, SAS
4. yes, AAA
5. yes, SSS
6. 16,28
Lesson 1 Exercises:
A.1. R∠ B. 1. Correct C. 1. a=6
2. E∠ 2. correct b=1
3. L∠ 3. Not correct
4. RE 4. correct 2. NP=24
5. YE 5. correct NO=12
6. 5 6. correct PO = 18
7. 3 7. DE, EF, DF
8. RS, ST, RT
9. RS, RT, ST
10. MO, MN, NO
Lesson 2
A.1. Similar, SSS B.1. Similar, SAS C.1. yes
2. Similar, SAS 2. Similar, AA 2. no
3. Similar, SAS 3. Similar, SAS, AA 3. no
4. Similar, SSS 4. Similar, AA 4. yes
5. Similar, AAA 5. Similar, SSS 5
6. Similar, SSS 6
7. Similar, SAS 7. BD = 5.7’
8. not similar 8. DC = 7.5
9. Similar, SAS 9
10.Similar, SSS 10
18, 22.5
Equilateral ∆
is equiangular
by AAA. Any
2 equilateral ∆
are similar.

Lesson 3
Exercises
1. 372in.
2. 15, 36
3. 6cm., 12cm., 14cm.
4. 28m.
5. 152cm.
What have your learned?
1. Similar, SAS
2. Similar, SAS
3. not similar
4. 4
5. 15
6-7. 6.3m
8 - 10. 66m.

Fun with Math
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