Section 5-5 Inequalities in Triangles SPI 22C: apply the Triangle Inequality Property to determine which sets of side lengths determine a triangleSPI 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

6 = 2 + 4, with c = 4, then 6 > 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.

Given: 1 is an exterior angle

Prove: m1 > m2 and m1 > m3

Proof: By the Exterior Angle Theorem , m1 = m2 + m3. Since the m2 > 0 and the m3 > 0, you can apply the Comparison Property of Inequality and conclude that m1 > m2 and m1 > m3.

Write a paragraph proof given the following information.

1 2

3

Applying the Corollary

x = 45 + 72x = 117

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.)

So, an exterior angle at P measures 117º.

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 mY > mZ.

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 mA > mB, then BC > AC.

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.

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.

Apply Properties of Triangles

Can a triangle have sides with the given lengths? Explain.

a. 2 m, 7 m, and 9 m

b. 4 yd, 6 yd, and 9 yd

NO2 + 7 is not greater than 9

YES4 + 6 > 96 + 9 > 44 + 9 > 6