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12/3/2015 Splash Screen 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 5-Minute Check 5 1 12/3/2015 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. Then/Now • Definition of Inequality –For any real numbers a and b, a>b if and only if there is a positive number c such that a=b+c –Ex: 5=2+3, then 5>2 and 5>3 Concept 2 12/3/2015 • In every triangle, the angle opposite the largest side is the largest angle • The angle opposite the smallest side is the smallest angle Concept 3 12/3/2015 Order Triangle Angle Measures List the angles of ΔABC in order from smallest to largest. The sides from the shortest to longest are AB, BC, and AC. The angles opposite these sides are C, A, and B, respectively. So, according to the Angle-Side Relationship, the angles from smallest to largest are C, A, B. Answer: C, A, B Example 2 Order Triangle Side Lengths List the sides of ΔABC in order from shortest to longest. Answer: AC, AB, BC Example 3 4 12/3/2015 List the angles of ΔTVX in order from smallest to largest. A. X, T, V B. X, V, T C. V, T, X D. T, V, X Example 2 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? Example 4 5 12/3/2015 List the sides of ΔRST in order from shortest to longest. A. RS, RT, ST B. RT, RS, ST C. ST, RS, RT D. RS, ST, RT Example 3 Concept 6 12/3/2015 Concept Use the Exterior Angle Inequality Theorem Example 1 7 12/3/2015 Use the Exterior Angle Inequality Theorem By the Exterior Angle Inequality Theorem, m14 > m4 and m14 > m11. In addition, m14 > m2 and m14 > m4 + m3, so m14 > m4 and m14 > m3. Since 11 and 9 are vertical angles, they have equal measure, so m14 > m9. m9 > m6 and m9 > m7, so m14 > m6 and m14 > m7. Example 1 Use the Exterior Angle Inequality Theorem Example 1 8 12/3/2015 Use the Exterior Angle Inequality Theorem By the Exterior Angle Inequality Theorem, m10 > m5 and m16 > m10, so m16 > m5. Since 10 and 12 are vertical angles, m12 > m5. m15 > m12, so m15 > m5. In addition, m17 > m5 + m6, so m17 > m5. Example 1 A. B. C. D. Example 1 9 12/3/2015 A. B. C. D. Example 1 Homework: Pg. 348 #’s 8-18 EVENS 10