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**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

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**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: m1 > m2 and m1 > m3 3 1 2 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.

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**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.

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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.

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**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.

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**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.

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**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

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4.7 Triangle Inequalities. In any triangle… The LARGEST SIDE lies opposite the LARGEST ANGLE. The SMALLEST SIDE lies opposite the SMALLEST ANGLE.

4.7 Triangle Inequalities. In any triangle… The LARGEST SIDE lies opposite the LARGEST ANGLE. The SMALLEST SIDE lies opposite the SMALLEST ANGLE.

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