Transcript Similarity Theorems

SIMILARITY
THEOREMS
Similarity in Triangles
Angle-Angle Similarity Postulate (AA~)If two angles of one triangle are
congruent to two angles of another
triangle, then the triangles are similar.
W
S
45
R
V
45
B
WRS  BVS
because of the AA~
Postulate.
Similarity in Triangles
Side-Angle-Side Similarity Postulate
(SAS~)- If an angle of one triangle is
congruent to an angle of a second
triangle, and the sides including the
angles are proportional, then the
triangles are similar.
16
C
T
32
32
28
12
E
U
P
21
TEA  CUP
because of the
A
SAS~ Postulate.
The scale factor is 4:3.
Similarity in Triangles
Side-Side-Side Similarity Postulate
(SSS~)- If the corresponding sides of two
triangles are proportional, then the
triangles are similar.
A
30
15
B
C
Q20
ABC  QRS
3
because of the
R
S
4
SSS~ Postulate.
The scale factor is 1:5.
6
Are the following triangles similar?
If so, what similarity statement can
be made. Name the postulate or
theorem you used.
F
J
H
G
K
Yes, FGH  KJH because
of the AA~ Postulate
Are the following triangles similar?
If so, what similarity statement can
be made. Name the postulate or
theorem you used.
M
G
3
6
O
H
R
10
No, these are not similar
because
4
I
Are the following triangles similar?
If so, what similarity statement can
be made. Name the postulate or
theorem you used.
A
20
X
25
25
Y
30
B
No, these are not similar
because
C
Are the following triangles similar?
If so, what similarity statement can
be made. Name the postulate or
theorem you used.
A
2
3
P
5
B
J
3
3
8
C
Yes, APJ  ABC because of
the SSS~ Postulate.
Explain why these triangles are
similar. Then find the value of x.
4.5
5
x
3
These 2 triangles are similar
because of the AA~ Postulate.
x=7.5
Explain why these triangles are
similar. Then find the value of x.
5
x
110
70
3
3
These 2 triangles are similar
because of the AA~ Postulate.
x=2.5
Explain why these triangles are
similar. Then find the value of x.
x
24
14
22
These 2 triangles are similar
because of the AA~ Postulate.
x=12
Explain why these triangles are
similar. Then find the value of x.
6
9
2
x
These 2 triangles are similar
because of the AA~ Postulate.
x= 12
Explain why these triangles are
similar. Then find the value of x.
4
5
x
15
These 2 triangles are similar
because of the AA~ Postulate.
x=8
Explain why these triangles are
similar. Then find the value of x.
x
7.5
12
18
These 2 triangles are similar
because of the AA~ Postulate.
x= 15
Similarity in Triangles
Side Splitter Theorem - If a line is parallel
to one side of a triangle and intersects the
other two sides, then it divides those sides
You can either
proportionally.
use
T
x
or
S
16
R
5
U
10
V
Theorem
Triangle Angle Bisector Theorem -If a
ray bisects an angle of a triangle, then it
divides the opposite side on the triangle
into two segments that are proportional to
the other two sides of the triangle.
A
C
D
B