Basic Mathematics for College Students

SECTION 9.5 Congruent Triangles and Similar Triangles

If two triangles have the same size and the same

shape, they are congruent triangles.

Corresponding partsof congruent triangles are

congruent (have the same measure).

DEFINITIONS AND CONCEPTS EXAMPLES

ABC  DEF

A B

C

D E

F

 

There are six pairs of congruent parts: three pairs of congruent

angles and three pairs of congruent sides.

••

••

••m(C)m(F) m(AB)m(DE)

m(B)m(E) m(AC)m(DF)

m(A)m(D) m(BC)m(EF)

822 Chapter 9 An Introduction to Geometry

Three ways to show that two triangles are

congruent are:

1.The SSS property If three sides of one triangle

are congruent to three sides of a second triangle,

the triangles are congruent.

N

O

M

6 in. 4 in. 4 in. 6 in.

7 in.

R

T

S

7 in.

2.The SAS property If two sides and the angle

between them in one triangle are congruent,

respectively, to two sides and the angle between

them in a second triangle, the triangles are

congruent.

DEF  XYZby the SAS property.

EZ X

Y

F

D

2 ft

5 ft

5 ft 2 ft

92° 92°

3.The ASA property If two angles and the side

between them in one triangle are congruent,

respectively, to two angles and the side between

them in a second triangle, the triangles are

congruent.

ABC  TUVby the ASA property.

A B

C

10 m

135° 20°

U

T

V

10 m

20°

135°

Similar triangleshave the same shape, but not

necessarily the same size.

We read the symbol as “is similar to.”

AAA similarity theorem

If the angles of one triangle are congruent to

corresponding angles of another triangle, the

triangles are similar.



EFGWXYby the AAA similarity theorem.

E F W X

G

Y

25°

15°

15°

140°

25°

140°

NO  ST

MN  RS

MO  RT

MNO  RSTby the SSS property.

DE  XY

D  X

DF  XZ

B  U

AB  TU

A  T

G  Y

F  X

E  W