Basic Mathematics for College Students

EXAMPLE (^4) Refer to the figure below. If , name the
congruent angles and the sides that are proportional.

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The formal definition of similar triangles requires that we establish a

correspondence between the vertices of the triangles. The definition also involves the

word proportional.

Recall that a proportionis a mathematical statement that two ratios (fractions)

are equal. An example of a proportion is

In this case, we say that and are 12 48 proportional.

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758 Chapter 9 An Introduction to Geometry

Similar Triangles

Two triangles are similar if and only if their vertices can be matched so that

corresponding angles are congruent and the lengths of corresponding sides

are proportional.

Self Check 4

If , name the

congruent angles and the sides

that are proportional.

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StrategyWe will establish the correspondence between the vertices of

and the vertices of.

WHYThis will, in turn, establish a correspondence between the congruent

corresponding angles and proportional sides of the triangles.

SolutionWhen we write , a correspondence between the vertices

of the triangles is established.

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Since the triangles are similar, corresponding angles are congruent:

The lengths of the corresponding sides are proportional. To simplify the notation,

we will now let , , , and so on.

Written in a more compact way, we have

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PQ

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PQm(PQ)CDm(CD)QRm(QR)

P  C Q  D R  E

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Property of Similar Triangles

If two triangles are similar, all pairs of corresponding sides are in proportion.

Now TryProblem 39

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