Basic Mathematics for College Students

One way to determine whether two triangles are congruent is to see if one triangle

can be moved onto the other triangle in such a way that it fits exactly. When we write

, we are showing how the vertices of one triangle are matched to the

vertices of the other triangle to obtain a “perfect fit.” We call this matching of points a

correspondence.

ABC  DEF

754 Chapter 9 An Introduction to Geometry

SECTION 9.5

Congruent Triangles and Similar Triangles

Objectives

1 Identify corresponding parts of

congruent triangles.

2 Use congruence properties to

prove that two triangles are

congruent.

3 Determine whether two

triangles are similar.

4 Use similar triangles to find

unknown lengths in application

problems.

In our everyday lives, we see many types of triangles. Triangular-shaped kites, sails, roofs,

tortilla chips, and ramps are just a few examples. In this section, we will discuss how to

compare the size and shape of two given triangles. From this comparison, we can make

observations about their side lengths and angle measures.

1 Identify corresponding parts of congruent triangles.

Simply put, two geometric figures are congruent if they have the same shape and size.

For example, if and shown below are congruent, we can write

ABC  DEF Read as “Triangle ABCis congruent to triangle DEF.”

ABC DEF

© iStockphoto.com/Lucinda Deitman

A

C

B D

F

E

ABC  DEF

 

Read as “Point Acorresponds to point D.”

Read as “Point Bcorresponds to point E.”

Read as “Point Ccorresponds to point F.”

When we establish a correspondence between the vertices of two congruent

triangles, we also establish a correspondence between the angles and the sides of the

triangles. Corresponding angles and corresponding sides of congruent triangles are

called corresponding parts.Corresponding parts of congruent triangles are always

congruent.That is, corresponding parts of congruent triangles always have the same

measure. For the congruent triangles shown above, we have

m(BC)m(EF) m(AC)m(DF) m(AB)m(DE)

m(A)m(D) m(B)m(E) m(C)m(F)

C 4 F

B 4 E

A 4 D

Congruent Triangles

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

the corresponding sides and the corresponding angles are congruent.

X

ZR

YQ P

5 in. 11 in.

27 ° 88 °

EXAMPLE (^1) Refer to the figure below, where.
a.Name the six congruent corresponding parts
of the triangles.
b.Find.
c.Find .m(XZ)
m(P)

XYZ  PQR