1. 6-4 Inequalities for One Triangle

2. Angle and Side Inequalities of a triangle (already discussed)
In the triangle below we can deduce that angle C is congruent to angle B.
In the following triangle, what can you deduce? A 15 15 B C A 13 15 B C

3. 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

4. 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

5. 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

6. Triangle Inequality Theorem
The sum of the lengths of any 2 sides of a triangle is greater than the length of the third side.
So given ∆ABC
AB +BC > AC
AB +AC > BC
AC + BC > AB C A B
If this rule is violated then the triangle with those side lengths cannot be created

7. 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

8. 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.

9. Homework PG