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
no
14  8  25
no
16  7  23
b) 16, 7, 23
c) 18, 8, 24
yes 18  8  24
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
6
20
0
16
_____ < x < _____
b) 8, 8
_____ < x < _____
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.
C ,A, B
b) List the sides from largest to
smallest.
35
DF , DE , EF