Download SAS Similarity Theorem

Geometry
7.5
Theorems for
Similar Triangles
Two Triangles can be proved
similar by using:
•Definition of similar polygons
•All angles congruent
•All sides proportional
•AA Postulate (2 angles = 2 angles)
Today we learn 2 additional methods:
•SAS Similarity Theorem
•SSS Similarity Theorem
SAS Similarity Theorem
If an angle of one triangle is congruent to
an angle of another triangle and the sides
including those angles are in proportion,
then the triangles are similar.
D
A
3
B
4
C
6
E
∆ABC ~ ∆DEF
8
m<A = m<D
F
small
big
3
6
4
=
8
SSS Similarity Theorem
If the sides of two triangles are in
proportion, then the triangles are similar.
∆ABC ~ ∆DEF
D
A
6
4
B
8
C
6
E
9
12
small
big
F
4
6
6
=
9
8
=
12
4.
3.
2.
Problems: State the Method and Similarity Statement
1.
2.
A
D
B
A
E
Method: _______
80
E
5
C
D
3
B
C
Statement : ________
L
3.
10
6
80
Method: _______
Statement : ________
4.
3
R
5
M
K
6
6
F
10
N
H
10
6
O
16
20
G
15
X
24
32
S
Method: _______
Method: _______
Statement : ________
Statement : ________
5.
6.
R
60
F
H
70 40
G
X
Q
10.
5
9
70
P
S
Method: _______
Statement : ________
T
7
80
S
6
R
Method: _______
Statement : ________
1.
Given:
AX BX

XC XD
A
B
X
Prove:
AB || DC
D
C
2.
Given:
AX AY

AB AC
Prove:
AY XY

AC BC
Y
C
A
X
B
Homework
pg. 264 CE #1-6
WE #1-13
Similarity Chart
All Polygons
•Definition:
•All angles congruent
•All sides proportional
Triangles
•AA Postulate (2 <‘s = 2 <‘s)
•SAS Similarity Theorem
•SSS Similarity Theorem
Properties of Similar ∆’s
• Similarity has some of the same
properties as equality and congruence.
• These properties include:
REFLEXIVE
SYMMETRIC
TRANSITIVE
Name 2 similar ∆’s. Justify with a theorem.
B
A
6 C
9
10
E
15
D
∆ABC ~ ∆DEC
by SAS Similarity
∆FHG ~ ∆XRS
by SSS Similarity
F
10
H
16
20
15
G
X
R
24
32
S
Name 2 similar ∆’s. Justify with a theorem
or postulate.
A
D
B
80˚
80˚
E
C
∆CDE ~ ∆CAB
by SAS Similarity
∆ADE ~ ∆ABC
by AA Postulate
C
6
D
3
A
10
E
5
B
From the homework
pg. 266 #3, 12