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The definition of inequality and the properties of inequalities can be applied to the measures of angles and segments, since these are real numbers. Consider 1, 2, and 3 in the figure shown. By the Exterior Angle Theorem, you know that m1 = m2 + m3. Since the angle measures are positive numbers, we can also say that m1 > m2 and by the definition of inequality. m1 > m3 Use the Exterior Angle Inequality Theorem Use the Exterior Angle Inequality Theorem The longest side and largest angle of ∆ABC are opposite each other. Likewise, the shortest side and smallest angle are opposite each other. Order Triangle Angle Measures List the angles of ΔABC in order from smallest to largest. Order Triangle Side Lengths List the sides of ΔABC in order from shortest to longest. Angle-Side Relationships HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie? Five-Minute Check (over Lesson 5–2) CCSS Then/Now Key Concept: Definition of Inequality Key Concept: Properties of Inequality for Real Numbers Theorem 5.8: Exterior Angle Inequality Example 1: Use the Exterior Angle Inequality Theorem Theorems: Angle-Side Relationships in Triangles Example 2: Order Triangle Angle Measures Example 3: Order Triangle Side Lengths Example 4: Real-World Example: Angle-Side Relationships Over Lesson 5–2 Find the coordinates of the centroid of the triangle with vertices D(–2, 9), E(3, 6), and F(–7, 0). A. (–4, 5) B. (–3, 4) C. (–2, 5) D. (–1, 4) Over Lesson 5–2 Find the coordinates of the orthocenter of the triangle with vertices F(–1, 5), G(4, 4), and H(1, 1). A. B. C. (2, 3) D. Over Lesson 5–2 ___ ___ In ΔRST, RU is an altitude and SV is a median. Find y if mRUS = 7y + 27. A. 5 B. 7 C. 9 D. 11 Over Lesson 5–2 ___ ___ In ΔRST, RU is an altitude and SV is a median. Find RV if RV = 6a + 3 and RT = 10a + 14. A. 3 B. 4 C. 21 D. 27 Over Lesson 5–2 Which of the following points is the center of gravity of a triangle? A. centroid B. circumcenter C. incenter D. orthocenter Content Standards G.CO.10 Prove theorems about triangles. Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others. You found the relationship between the angle measures of a triangle. • Recognize and apply properties of inequalities to the measures of the angles of a triangle. • Recognize and apply properties of inequalities to the relationships between the angles and sides of a triangle.