8.3 Methods of Proving
Triangles Similar
Objective:
After studying this section, you will be able to use
several methods to prove triangles are similar.
Postulate
If there exists a
correspondence between
the vertices of two triangles
such that the three angles
of one triangle are
congruent to the three
angles of another triangle,
then the triangles are
similar. (AAA)
Theorem
If there exists a
correspondence between
the vertices of two triangles
such that the two angles
of one triangle are
congruent to the two
corresponding angles of
other, then the triangles are
similar. (AA)
Given: A  D
B  E
A
Prove: ABC ~ DEF
D
E
B
F
C
Theorem
If there exists a
correspondence between
the vertices of two triangles
such that the ratios of the
measures of corresponding
sides are equal, then the
triangles are similar.
(SSS similarity)
A
Given: AB  BC  AC
DE EF DF
Prove: ABC ~ DEF
D
E
B
F
C
Theorem
If there exists a
correspondence between
the vertices of two triangles
such that the ratios of the
measures of two pairs of
corresponding sides are
equal and the included
angles are congruent, then
the triangles are similar.
(SAS Similarity)
AB BC
Given: DE  EF
B  E
Prove: ABC ~ DEF
A
D
E
B
F
C
Given: parallelogram YSTW
D
C
Prove: BFE ~ CFD
F
A
B
E
Given:
LP  EA
L
N is themidpointof LP
N
P and R trisect EA
Prove: PEN ~ PAL
E
P
R
A
Given: KH is thealtitude to
J
hypotenuseGJ of right GHJ
Prove: KHJ ~ HGJ
G
K
H
The sides of one triangle are 8, 14, and
12, and the sides of another triangle are
18, 21, and 12. Prove that the triangles
are similar.
Summary: If you were to draw
a triangle and have a line
parallel to the base,creating
two triangles, would the
triangles be similar if you had
a right triangle? Obtuse?
Acute? Why?
Homework: worksheet