Basic Mathematics for College Students

9.3 Triangles 739

The Language of Mathematics Tick markscan be used to denote the sides of

a triangle that have the same length. They can also be used to indicate the angles

of a triangle with the same measure. For example, we can show that the base

angles of the isosceles triangle below are congruent by using single tick marks.

Dis opposite , and Eis opposite

. By the isosceles triangle theorem,
if m(FD)m(FE), then m(D) m(E).

FD

FE

F

D E

Converse of the Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite the angles

have the same length, and the triangle is isosceles.

EXAMPLE (^2) Is the triangle shown here an isosceles triangle?
StrategyWe will consider the measures of the angles of the triangle.
WHYIf two angles of a triangle are congruent, then the
sides opposite the angles have the same length, and the
triangle is isosceles.
Solution and have the same measure, 50°.
By the converse of the isosceles triangle theorem, if
m(A)m(B), we know that m(BC)m(AC)and that ABCis isosceles.

A B

Self Check 2

Is the triangle shown below an

isosceles triangle?

Now TryProblems 33 and 35

50 °

80 °

A B

C

50 °

78 °

23 °

79 °

4 Find unknown angle measures of triangles.

If you draw several triangles and carefully measure each angle with a protractor, you
will find that the sum of the angle measures of each triangle is 180°. Two examples are
shown below.

62 ° 29 °

89 °

62 ° + 89° + 29° = 180°

37 ° 110 °

33 °

37 ° + 110° + 33° = 180°

Another way to show this important fact about the sum of the angle measures of
a triangle is discussed in Problem 82 of the Study Set at the end of this section.

Angles of a Triangle

The sum of the angle measures of any triangle is 180°.

If a mathematical statement is written in the form if p ...,then q.. ., we call the
statement if q... , then p.. .its converse.The converses of some statements are true,
while the converses of other statements are false. It is interesting to note that the
converse of the isosceles triangle theorem is true.