6-4 Inequalities for One
Triangle
Angle and Side Inequalities of a triangle (already
discussed)
 In the triangle below we can deduce that angle C is
congruent to angle B.
A
15
15
B
C
 In the following triangle, what can you deduce?
A
15
B
13
C
Theorem
 If one side of a triangle is longer than the second
side, then the angle opposite the first side is larger
than the angle opposite the second side.
 WE SHOULD ALREADY KNOW THIS
Theorem
 If one angle of a triangle is larger than a second
angle, then the side opposite the first angle is longer
than the side opposite the second angle.
 WE SHOULD ALREADY KNOW THIS AS WELL
2 Corollaries
 The perpendicular segment from a point to a line is
the shortest segment from the point to the line.

This we should know because of how we measure distance.
 The perpendicular segment from a point to a plane is
the shortest segment from the point to the plane.
 Smartboard demo
Triangle Inequality Theorem
 The sum of the lengths of any 2 sides of a triangle is
greater than the length of the third side.
C
 So given ∆ABC
 AB +BC > AC
 AB +AC > BC
 AC + BC > AB
A
B
If this rule is violated then the triangle with those side lengths cannot be created
Is it possible?
 Is it possible to have a triangle with side lengths of…
 5, 8, 12

3, 5, 8

50, 4, 7

9, 21, 92

4, 10, 15
 If you are provided with 2 sides and are asked to find
the range of values that are acceptable for the third
side you simply subtract the 2 given number for the
lower bound and add the two sides to get the upper
bound for the possible values of the 3rd side.
 Sides 5 and 12. What is the range for the third side.
 Sides 23 and 32. What is the range of the third side.
Homework
PG. 222 1-16