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7-3 Triangle Similarity
I CAN
-Use the triangle similarity theorems to
determine if two triangles are similar.
-Use proportions in similar triangles to solve
for missing sides.
Recall in 7-2, to prove that two polygons
are similar you had to:
Show all corresponding angles are congruent
AND
Show all corresponding sides are proportional
Triangle Similarity Theorems are “shortcuts”
for showing two triangles are similar.
Example
D
9
B
E
12
10
C
A
18
6
5
F
Similarity Statement
EFD
ABC ~
Reason
by AA
Because A  E and C  D
justification
Example
Explain why the triangles
are similar and write a
similarity statement.
By the Triangle Sum Theorem, mC = 47°, so C  F.
B  E by the Right Angle Congruence Theorem.
Therefore, ∆ABC ~ ∆DEF by AA ~.
Example
Verify that the triangles are similar.
∆PQR and ∆STU
Therefore ∆PQR ~ ∆STU by SSS ~.
Example
Verify that the triangles are similar.
∆DEF and ∆HJK
D  H by the Definition of Congruent Angles.
Therefore ∆DEF ~ ∆HJK by SAS ~.
Example
Verify that ∆TXU ~ ∆VXW.
TXU  VXW by the
Vertical Angles Theorem.
Therefore ∆TXU ~ ∆VXW by SAS ~.
Your Turn
Your Turn
Your Turn
Find the value of x such that ∆ACE ~ ∆BCD
C
Why is ∆ACE ~ ∆BCD?
C
3
12
B
3
D
B
x
28
A
3
x+3
=
12
C
12
28
12(x + 3) = 84
12x + 36 = 84
– 36 – 36
12x = 48
x=4
E
D
x+3
A
E
28
Example
Explain why ∆ABE ~ ∆ACD, and
then find CD.
Step 1 Prove triangles are similar.
A  A by Reflexive Property of , and B  C
since they are both right angles.
Therefore ∆ABE ~ ∆ACD by AA ~.
Step 2 Find CD.
x(9) = 5(3 + 9)
9x = 60
Example
Explain why ∆RSV ~ ∆RTU
and then find RT.
Step 1 Prove triangles are similar.
It is given that S  T.
R  R by Reflexive Property of .
Therefore ∆RSV ~ ∆RTU by AA ~.
Step 2 Find RT.
RT(8) = 10(12)
8RT = 120
RT = 15
Your Turn
Properties of Similar Triangles
7-4
 I can use the triangle proportionality
theorem and its converse.
 I can set up and solve problems using
properties of similar triangles.
Example: AF = 3, FC = 5, AE = 2. Find BE
2 3

x 5
3x  10
10
x
3
OR
2 x

3 5
3x  10
10
x
3
Example
Find US.
It is given that
, so
by
the Triangle Proportionality Theorem.
US(10) = 56
Example
Find PN.
2PN = 15
PN = 7.5
Your Turn
Solve for x.
Example
Solve for x.
Example
Example
I
Solve for x, y, and w.
4
x
N
y
w
E
4 x

2 3
2x  12
5
L
A
3
2
D
IN
EI

AD LS
12
S
x6
Example
I
Solve for x, y, and w.
4
x=6
N
y
w
AN EL

DA LS
E
L
A
3
2
D
OR
y 5

2 3
3 y  10
5
S
12
y 5

4 6
OR
4 6

y 5
10
y
3
Example
I
Solve for x, y, and w.
4
x=6
N
10
y
3
w
E
6
w

14 12
5
L
A
3
2
D
IE NE

IS DS
12
14w  72
S
w  5.14
Example
BD = 8; DF = 6; CE = 16.
EG = ________
Example
BD = 2x – 2; DF = 4; CE = x + 2; EG = 8
Find BD and CE
Your Turn
Example: SHOW EF BC if BE = 21,
AE = 42, CF = 15, and AF = 30
21 ? 15

42 30
1 1

2 2
So, EF
BC
Example
Verify that
Since
.
,
by the Converse of the
Triangle Proportionality Theorem.
Your Turn
AC = 36 cm, and BC = 27 cm.
Verify that
Since
.
,
by the Converse of the
Triangle Proportionality Theorem.
The previous theorems and corollary lead to the
following conclusion.
Example
Find PS and SR.
by the ∆  Bisector Theorem.
40(x – 2) = 32(x + 5)
PS = x – 2
= 30 – 2 = 28
40x – 80 = 32x + 160
40x – 80 = 32x + 160
8x = 240
x = 30
SR = x + 5
= 30 + 5 = 35
Example
Find AC and DC.
by the ∆  Bisector Theorem.
4y = 4.5y – 9
–0.5y = –9
y = 18
So DC = 9 and AC = 16.
Your Turn