Presentation on theme: "Section 5-5 Inequalities in Triangles SPI 22C: apply the Triangle Inequality Property to determine which sets of side lengths determine a triangle SPI."— Presentation transcript:
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 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: m 1 > m 2 and m 1 > m 3 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. Write a paragraph proof given the following information
Applying the Corollary x = x = 117 In PQR, m
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. a. 2 m, 7 m, and 9 m b. 4 yd, 6 yd, and 9 yd NO is not greater than 9 YES > > > 6