Download Theorem 5.10 Triangle Sides Inequality Theorem (TSIT)

Geometry Section 5.5
Triangle Inequalities
Theorem 5.12 Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle
will be greater than the length of the third side.
AB  BC  AC
AB  AC  BC
AC  BC  AB
Examples: Which of the following are possible
lengths for the sides of a triangle?
a) 14, 8, 25
NO 8  14  25
b) 16, 7, 23
NO 7  16  23
c) 18, 8, 24
YES 8  18  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.
To get the lower bounds...
a) 7, 13
6
20
_____ < x < _____
subtract
To get the upper bounds...
b) 8, 8
0
16
_____ < x < _____
add
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.
Theorem 5.10
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 largest angle will be opposite the longest side.
The converse of this theorem is also true.
Theorem 5.11
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 longest side will be opposite the largest 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