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Triangle Angle-Sum Theorem & Exterior Angle Theorem Exploration

Explore the Triangle Angle-Sum Theorem and Exterior Angle Theorem with proofs, examples, and real-world applications. Understand corollaries, classify triangles, and solve angle measures in right triangles.

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Triangle Angle-Sum Theorem & Exterior Angle Theorem Exploration

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 4–1) CCSS Then/Now New Vocabulary Theorem 4.1: Triangle Angle-Sum Theorem Proof: Triangle Angle-Sum Theorem Example 1: Real-World Example: Use the Triangle Angle-Sum Theorem Theorem 4.2: Exterior Angle Theorem Proof: Exterior Angle Theorem Example 2: Real-World Example: Use the Exterior Angle Theorem Corollaries: Triangle Angle-Sum Corollaries Example 3: Find Angle Measures in Right Triangles Lesson Menu

  3. Classify ΔRST . A. acute B. equiangular C. obtuse D. right 5-Minute Check 1

  4. Find y if ΔRST is an isosceles triangle withRS  RT. ___ ___ A. 8 B. 10 C. 12 D. 14 5-Minute Check 2

  5. Find x if ΔABC is an equilateral triangle. A. 2 B. 4 C. 6 D. 8 5-Minute Check 3

  6. A.ΔABC B.ΔACB C.ΔADC D.ΔCAB 5-Minute Check 4

  7. Classify ΔMNO as scalene, isosceles, or equilateral if MN = 12, NO = 9, and MO = 15. A. scalene B. isosceles C. equilateral 5-Minute Check 5

  8. Which is not a classification for ΔFGH? A. acute B. scalene C. isosceles D. equiangular 5-Minute Check 6

  9. 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

  10. You classified triangles by their side or angle measures. • Apply the Triangle Angle-Sum Theorem. • Apply the Exterior Angle Theorem. Then/Now

  11. auxiliary line • exterior angle • remote interior angles • flow proof • corollary Vocabulary

  12. Concept 1

  13. Concept 2

  14. Use the Triangle Angle-Sum Theorem SOFTBALL The diagram shows the path of the softball in a drill developed by four players. Find the measure of each numbered angle. Understand Examine the information in the diagram. You know the measures of two angles of one triangle and only one measure of another. You also know that 1 and 2 are vertical angles. Example 1

  15. Use the Triangle Angle-Sum Theorem Plan Find m1 first because the measure of two angles of the triangle are known. Use the Vertical Angles Theorem to find m2. Then you will have enough information to find the measure of 3. Solve Triangle Angle-Sum Theorem Simplify. Subtract 117 from each side. Example 1

  16. Use the Triangle Angle-Sum Theorem 1 and 2 are congruent vertical angles. So, m2 = 63. Triangle Angle-Sum Theorem Simplify. Subtract 142 from each side. Answer: Therefore, m1 = 63, m2 = 63, and m3 = 38. CheckThe sums of the measures of the angles in each triangle should be 180.m1 + 43 + 74 = 63 + 43 + 74 or 180m2 + m3 + 79 = 63 + 38 + 79 or 180 Example 1

  17. Find the measure of 3. A. 95 B. 75 C. 57 D. 85 Example 1

  18. Concept 3

  19. Concept 4

  20. Use the Exterior Angle Theorem GARDENING Find the measure of FLW in the fenced flower garden shown. mLOW + mOWL = mFLW Exterior Angle Theorem x + 32 = 2x – 48 Substitution 32 = x – 48 Subtract x from each side. 80 = x Add 48 to each side. Answer: So, mFLW = 2(80) – 48 or 112. Example 2

  21. The piece of quilt fabric is in the shape of a right triangle. Find the measure of ACD. A. 30 B. 40 C. 50 D. 130 Example 2

  22. Concept 5

  23. m1 = 48 + 56 Exterior Angle Theorem = 104 Simplify. If 2 s form a linear pair, they are supplementary. 104 + m2 = 180 Substitution 76 Subtract 104 from each side. Find Angle Measures in Right Triangles Find the measure of each numbered angle. Example 3

  24. m 3 = 90 – 48 If 2 s form a right angle, they are complementary. = 42 Simplify. Triangle Angle-Sum Theorem (90 – 34) + m2+ m 4 = 180 56 + 76 + m 4 = 180 Substitution Simplify. 132 + m4 = 180 48 Subtract 132 from each side. Find Angle Measures in Right Triangles Example 3

  25. m5 + 41 + 90 = 180 Triangle Angle-Sum Theorem m5 + 143 = 180 Simplify. 49 Subtract 131 from each side. m1 = 104,m2 = 76, m3 = 42,m4 = 48, m5 = 49 Find Angle Measures in Right Triangles Example 3

  26. Find m3. A. 50 B. 45 C. 85 D. 130 Example 3

  27. End of the Lesson

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