Basic Mathematics for College Students

9.5 Congruent Triangles and Similar Triangles 757

WHYThen we know that the two triangles are
congruent by the SAS property.

SolutionSince vertical angles are congruent,

From the figure, we see that

and

Since 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,
ABC  EDCby the SAS property.

AC  EC BC  DC

 1   2

Now TryProblem 35

A

B

C

D

E

Self Check 2

Are the triangles in the figure

below congruent? Explain why

or why not.

A

10 cm

1

2

10 cm

5 cm

5 cm

E

B

C

D

EXAMPLE (^3) Are and in the figure on the right congruent?
StrategyWe will show that 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.
WHYThen we know that the two triangles
are congruent by the ASA property.
SolutionFrom the markings on the figure,
we know that two pairs of angles are congruent.
These angles are marked with 1 tick mark, which
indicates that they have the same measure.
These angles are marked with 2 tick marks, which
indicates that they have the same measure.
From the figure, we see that the triangles have side in common. Furthermore,
is between each pair of congruent angles listed above. Since every segment is
congruent to itself, we also have
Knowing that two angles and the side between them in are congruent,
respectively, to two angles and the side between them in , we can conclude
that RST  RUTby the ASA property.

RUT

RST

RT  RT

RT

RT

STR  UTR

SRT  URT

RST RUT

Now TryProblem 37

R

T

QS

Self Check 3

Are the triangles in the following

figure congruent? Explain why or

why not.

R

S

T

U

3 Determine whether two triangles are similar.

We have seen that congruent triangles have the same shape and size.Similar triangles
have the same shape, but not necessarily the same size. That is, one triangle is an exact
scale model of the other triangle. If the triangles in the figure below are similar, we can
write ABCDEF(read the symbolas “is similar to”).

Success Tip Note that congruent triangles are always similar, but similar

triangles are not always congruent.

A

C

B D

F

E