Honors Geometry Section 4.8

Triangle Inequalities

Goals for today’s class:

1. Learn and be able to apply the Triangle Inequality Theorem, the Triangle Side Inequality Theorem and the Triangle Angle Inequality Theorem

Triangle Inequality Theorem ( IT)The sum of the lengths of any two sides of a triangle is greater than

the length of the third side.

Examples: Which of the following are possible lengths for the sides of a triangle?a) 14, 8, 25

b) 16, 7, 23

c) 18, 8, 24

25814 no

23716 no

24818 yes

Examples: The lengths of two sides of a triangle are given. Write a compound inequality (two inequalities in one) that expresses the possible values of x, the length of the third side.

a) 7, 13 _____ < x < _____

b) 8, 8 _____ < x < _____

6 20

0 16

The Isosceles Triangle Theorem states “If two sides of a triangle are

congruent, then the angles opposite them are congruent.” The following theorem covers the case where two sides of a triangle are

not congruent.

Triangle Sides Inequality Theorem (TSIT)

In a triangle, if two sides are not congruent, then the angles

opposite those sides are not congruent and the larger angle will

be opposite the longer side.

The converse of this theorem is also true.

Triangle Angles Inequality Theorem (TAIT)

In a triangle, if two angles are not congruent, then the sides opposite those angles are not

congruent and the longer side will be opposite the larger angle.

Examples: a) List the angles from smallest to largest.

B , , A C

b) List the sides from largest to smallest.

35 EF , DE , DF