Basic Mathematics for College Students

It is possible to conclude that two triangles are similar without having to show that
all three pairs of corresponding angles are congruent and that the lengths of all three
pairs of corresponding sides are proportional.

9.5 Congruent Triangles and Similar Triangles 759

AAA Similarity Theorem

If the angles of one triangle are congruent to corresponding angles of another

triangle, the triangles are similar.

EXAMPLE (^5) In the figure on the right,. Are and
similar triangles?
StrategyWe will show that the angles of
one triangle are congruent to corresponding
angles of another triangle.
WHYThen we know that the two triangles
are similar by the AAA property.
SolutionSince vertical angles are congruent,
This is one pair of congruent corresponding angles.
In the figure, we can view as a transversal cutting parallel line segments
and. Since alternate interior angles are then congruent, we have:
This is a second pair of congruent corresponding angles.
Furthermore, we can view as a transversal cutting parallel line segments
and. Since alternate interior angles are then congruent, we have:
This is a third pair of congruent corresponding angles.
These observations are summarized in
the figure on the right. We see that
corresponding angles of are
congruent to corresponding angles of

. By the AAA similarity theorem,
we can conclude that

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NQM

PQR

QRP  QMN

PR MN

RM

·

RPQ  MNQ

MN

PN PR

·

PQR  NQM

PRMN PQR NQM

Self Check 5

In the figure below,.

Are and similar

triangles?

XYA XZB

YAZB

Now TryProblems 41 and 43

X A B

Z

Y

P

R

Q

M

N

P

R

Q

M

N

Self Check 6

In the figure below,

. Find:
a.x b.y

DEFGHI

Now TryProblem 53

D

E

F

I

G

H

(^1518)
4.5
13.5
x
y
EXAMPLE (^6) In the figure below,. Find: a. b.
StrategyTo find , we will write a
proportion of corresponding sides so that
is the only unknown. Then we will solve
the proportion for. We will use a similar
method to find.
WHYSince , we know
that the lengths of corresponding sides of
and are proportional.
Solution
a.When we write , a correspondence between the vertices of the
two triangles is established.

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RST JKL

RSTJKL

y

x

x

x

RSTJKL x y

T

L

K

J

S

R

48 32 20

36

x

y

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