Download Similarity Theorems

SIMILARITY IN
TRIANGLES
&
THEOREMS
CCGPS STANDARDS

MCC9-12.G.SRT.2 Given two figures, use the
definition of similarity in terms of similarity
transformations to decide if they are similar;
explain using similarity transformations the
meaning of similarity for triangles as the
equality of all corresponding pairs of angles and
the proportionality of all corresponding pairs of
sides.


MCC9-12.G.SRT.3 Use the properties of
similarity transformations to establish the AA
criterion for two triangles to be similar.
CCGPS STANDARDS

MCC9-12.G.SRT.4 Prove theorems about
triangles. Theorems include: a line parallel to one
side of a triangle divides the other two
proportionally, and conversely; the Pythagorean
Theorem proved using triangle similarity.


MCC9-12.G.SRT.5 Use congruence and
similarity criteria for triangles to solve problems
and to prove relationships in geometric figures.
ESSENTIAL QUESTION
What
strategies can I use to
determine missing side lengths
and areas of similar figures?
How do I know which method to
use to prove two triangles
similar?
In geometry, two triangles
are similar when one is a
replica (scale model) of
the other.
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EXIT
Consider Dr. Evil and Mini Me from
Mike Meyers’ hit movie Austin Powers.
Mini Me is supposed to be an exact
replica of Dr. Evil.
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EXIT
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How do we know if
triangles are similar or
proportional?
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Triangles are similar (~)
if corresponding angles
are equal and
the ratios of the
lengths of corresponding
sides are equal.
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EXIT
Congruent Angles and Proportional Sides
A
D
E
B
C
ABC  DEF
F
Proportional sides have a constant ratio – known as a
scale factor or ratio of similitude.
A  D
B  E
C  F
AB
DE
=
BC
EF
=
AC
DF
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
Please complete the Ways to
Prove Triangles Similar
Worksheet.
Similarity in Triangles
Side Splitter Theorem
Triangle Proptionality Theorem - If a line
is parallel to one side of a triangle and
You can
either
intersects
the
other two sides, then it
use sides proportionally. T
divides those
x
or
S
16
R
5
U
10
V
T
x
S
16
R
5
U
10
V
Segment Addition
Postulate
TS + SR = TR
TU + UV = TV
Theorem
If three parallel lines intersect two
transversals, then the segments
intercepted are proportional.
c
d
a
b
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
Complete the practice sheets.