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Triangle Inequality Theorems & Applications

Understand triangle angle measures, order side lengths, angle-side relationships, real-world examples, and theorems. Practice finding centroids and orthocenters, and prove triangle theorems.

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Triangle Inequality Theorems & Applications

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  1. Splash Screen

  2. Concept

  3. Concept

  4. 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 Ð1, Ð2, and Ð3 in the figure shown. By the Exterior Angle Theorem, you know that mÐ1 = mÐ2 + mÐ3. Since the angle measures are positive numbers, we can also say that mÐ1 > mÐ2 and mÐ1 > mÐ3 by the definition of inequality. Concept

  5. Concept

  6. Use the Exterior Angle Inequality Theorem Example 1

  7. Use the Exterior Angle Inequality Theorem Example 1

  8. The longest side and largest angle of ∆ABC are opposite each other. Likewise, the shortest side and smallest angle are opposite each other. Concept

  9. Concept

  10. Order Triangle Angle Measures List the angles of ΔABC in order from smallest to largest. Example 2

  11. Order Triangle Side Lengths List the sides of ΔABC in order from shortest to longest. Example 3

  12. 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

  13. End of the Lesson

  14. 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 Lesson Menu

  15. 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) 5-Minute Check 1

  16. A. B. C.(2, 3) D. Find the coordinates of the orthocenter of the triangle with vertices F(–1, 5), G(4, 4), and H(1, 1). 5-Minute Check 2

  17. ___ ___ In ΔRST, RU is an altitude and SV is a median.Find y if mRUS = 7y + 27. A. 5 B. 7 C. 9 D. 11 5-Minute Check 3

  18. ___ ___ 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 5-Minute Check 4

  19. 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

  20. 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. CCSS

  21. 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

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