1. Angles of Triangles LESSON 4–2

2. Lesson Menu Five-Minute Check (over Lesson 4–1) TEKS 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

3. Over Lesson 4–1 5-Minute Check 1 A.acute B.equiangular C.obtuse D.right Classify ΔRST.

4. Over Lesson 4–1 5-Minute Check 2 A.8 B.10 C.12 D.14 Find y if ΔRST is an isosceles triangle with RS  RT. ___

5. Over Lesson 4–1 5-Minute Check 3 A.2 B.4 C.6 D.8 Find x if ΔABC is an equilateral triangle.

6. Over Lesson 4–1 5-Minute Check 4 A.ΔABC B.ΔACB C.ΔADC D.ΔCAB

7. Over Lesson 4–1 5-Minute Check 5 A.scalene B.isosceles C.equilateral Classify ΔMNO as scalene, isosceles, or equilateral if MN = 12, NO = 9, and MO = 15.

8. Over Lesson 4–1 5-Minute Check 6 A.acute B.scalene C.isosceles D.equiangular Which is not a classification for ΔFGH?

9. TEKS Targeted TEKS G.6(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. Mathematical Processes G.1(D), G.1(F)

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

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

12. Concept 1

13. Concept 2

14. Example 1 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. AnalyzeExamine 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.

15. Example 1 Use the Triangle Angle-Sum Theorem Triangle Angle-Sum Theorem Simplify. Subtract 117 from each side. FormulateFind 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. Determine

16. Example 1 Use the Triangle Angle-Sum Theorem Triangle Angle-Sum Theorem Simplify. Subtract 142 from each side.  1 and  2 are congruent vertical angles. So, m  2 = 63. Answer: Therefore, m  1 = 63, m  2 = 63, and m  3 = 38. JustifyThe 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

17. Example 1 Use the Triangle Angle-Sum Theorem EvaluateBy identifying each part of the problem, this complex problem could be separated into three manageable pieces. The properties of triangles were used to check the reasonableness of the answers found.

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

19. Concept 3

20. Concept 4

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

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

23. Concept 5

24. Example 3 Find Angle Measures in Right Triangles Find the measure of each numbered angle. If 2  s form a linear pair, they are supplementary. Exterior Angle Theorem m  1 = 38 + 32 Simplify. = 70 Substitution 70 + m  2 = 180 Subtract 70 from each side. 110

25. Example 3 Find Angle Measures in Right Triangles Subtract 78 from each side. 102 Simplify. 78+ m  4 = 180 If 3  s form a linear pair, they are supplementary 46+ 32+ m  4 = 180 Simplify. = 46 m  3 + 64 = 110 Exterior Angle Theorem

26. Example 3 Find Angle Measures in Right Triangles Subtract 143 from each side. 37 Simplify. m  5 + 143 = 180 Triangle Angle-Sum Theorem m  5 + 102+ 41 = 180 m  1 = 70, m  2 = 110, m  3 = 46, m  4 = 102, m  5 =37

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

28. Angles of Triangles LESSON 4–2