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

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.

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 = 45 + 72 x = 117 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 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.

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. 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