Presentation on theme: "Example 6 = 2 + 4, with c = 4, then 6 > 2"— Presentation transcript:
1Example 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 algebraicallyObjectives:use inequalities involving angles and sides of trianglesComparison Property of InequalityIf a = b + c and c > 0, then a > bExample6 = 2 + 4, with c = 4, then 6 > 2
2Using Property to Prove Corollary Corollary to the Triangle Exterior Angle TheoremThe 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 angleProve: m1 > m2 and m1 > m3312Proof: 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 thatm1 > m2 and m1 > m3.
3Applying 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 = 117So, an exterior angle at P measures 117º.
4Triangle PropertiesTheorem 5-10If 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.
5Real-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.
6Properties of Triangles Theorem 5-12 Triangle InequalityThe sum of the lengths of any two sides of a triangle is greater than the length of the third side.
7Apply Properties of Triangles Can a triangle have sides with the given lengths? Explain.NO2 + 7 is not greater than 9a. 2 m, 7 m, and 9 mb. 4 yd, 6 yd, and 9 ydYES4 + 6 > 96 + 9 > 44 + 9 > 6