1. Example 6 = 2 + 4, with c = 4, then 6 > 2
Section 5-5 Inequalities in Triangles SPI 22C: apply the Triangle Inequality Property to determine which sets of side lengths determine a triangle SPI 32E: solve problems involving congruent angles given angle measures expressed algebraically
Objectives:
use inequalities involving angles and sides of triangles
Comparison Property of Inequality
If a = b + c and c > 0, then a > b
Example
6 = 2 + 4, with c = 4, then 6 > 2

2. Using Property to Prove Corollary
Corollary to the Triangle Exterior Angle Theorem
The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles.
Write a paragraph proof given the following information.
Given: 1 is an exterior angle
Prove: m1 > m2 and m1 > m3 3 1 2
Proof: By the Exterior Angle Theorem , m1 = m2 + m3. Since the m2 > 0 and the m3 > 0, you can apply the Comparison Property of Inequality and conclude that
m1 > m2 and m1 > m3.

3. Applying the Corollary
In ∆ PQR, m<Q = 45º, and m<R = 72º.  Find the measure of an exterior angle at P.
It is always helpful to draw a diagram and label it with the given information.
                                               
Then, using the theorem set the exterior angle ( x ) equal to the sum of the two non-adjacent interior angles (45º and 72º.)
x = x = 117
So, an exterior angle at P measures 117º.

4. Triangle Properties
Theorem 5-10
If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.
If XZ > XY, then mY > mZ.
Theorem 5-11 (Converse of Thm 5-10)
If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.
If mA > mB, then BC > AC.

5. Real-World Connection
A landscaper is designing a triangular deck. She wants to place benches in the two larger corners. Which corners have the largest angles?
Angles B and C have the larger angles, since they are opposite the two longer sides.

6. Properties of Triangles
Theorem 5-12 Triangle Inequality
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

7. Apply Properties of Triangles
Can a triangle have sides with the given lengths? Explain. NO
2 + 7 is not greater than 9
a. 2 m, 7 m, and 9 m
b. 4 yd, 6 yd, and 9 yd
YES
4 + 6 > 9
6 + 9 > 4
4 + 9 > 6