1. Splash Screen

2. Lesson Menu Five-Minute Check (over Lesson 10–3) Then/Now New Vocabulary Theorem 10.6: Inscribed Angle Theorem Proof: Inscribed Angle Theorem (Case 1) Example 1: Use Inscribed Angles to Find Measures Theorem 10.7 Example 2: Use Inscribed Angles to Find Measures Example 3: Use Inscribed Angles in Proofs Theorem 10.8 Example 4: Find Angle Measures in Inscribed Triangles Theorem 10.9 Example 5: Real-World Example: Find Angle Measures

3. Over Lesson 10–3 5-Minute Check 1 A.60 B.70 C.80 D.90

4. Over Lesson 10–3 5-Minute Check 2 A.40 B.45 C.50 D.55

5. Over Lesson 10–3 5-Minute Check 3 A.40 B.45 C.50 D.55

6. Over Lesson 10–3 5-Minute Check 4 A.40 B.30 C.25 D.22.5

7. Over Lesson 10–3 5-Minute Check 5 A.24.6 B.26.8 C.28.4 D.30.2

8. Over Lesson 10–3 5-Minute Check 6 A. B. C. D.

9. Then/Now You found measures of interior angles of polygons. (Lesson 6–1) Find measures of inscribed angles. Find measures of angles of inscribed polygons.

10. Vocabulary inscribed angle intercepted arc

11. Concept

13. Example 1 Use Inscribed Angles to Find Measures A. Find m  X. Answer: m  X = 43

14. Example 1 Use Inscribed Angles to Find Measures B. = 2(52) or 104

15. Example 1 A.47 B.54 C.94 D.188 A. Find m  C.

16. Example 1 A.47 B.64 C.94 D.96 B.

17. Concept

18. Example 2 Use Inscribed Angles to Find Measures ALGEBRA Find m  R.  R   S  R and  S both intercept. m  R  m  SDefinition of congruent angles 12x – 13= 9x + 2Substitution x= 5Simplify. Answer: So, m  R = 12(5) – 13 or 47.

19. Example 2 A.4 B.25 C.41 D.49 ALGEBRA Find m  I.

20. Example 3 Use Inscribed Angles in Proofs Write a two-column proof. Given: Prove: ΔMNP  ΔLOP 1. Given Proof: StatementsReasons LO  MN2. If minor arcs are congruent, then corresponding chords are congruent.

21. Example 3 Use Inscribed Angles in Proofs Proof: StatementsReasons  M   L 4. Inscribed angles of the same arc are congruent.  MPN   OPL5. Vertical angles are congruent. ΔMNP  ΔLOP6. AAS Congruence Theorem 3. Definition of intercepted arc  M intercepts and  L intercepts.

22. Example 3 Write a two-column proof. Given: Prove: ΔABE  ΔDCE Select the appropriate reason that goes in the blank to complete the proof below. 1. Given Proof: StatementsReasons AB  DC2. If minor arcs are congruent, then corresponding chords are congruent.

23. Example 3 Proof: StatementsReasons  D   A 4.Inscribed angles of the same arc are congruent.  DEC   BEA5.Vertical angles are congruent. ΔDCE  ΔABE6. ____________________ 3. Definition of intercepted arc  D intercepts and  A intercepts.

24. Example 3 A.SSS Congruence Theorem B.AAS Congruence Theorem C.Definition of congruent triangles D.Definition of congruent arcs

25. Concept

26. Example 4 Find Angle Measures in Inscribed Triangles ALGEBRA Find m  B. ΔABC is a right triangle because  C inscribes a semicircle. m  A + m  B + m  C= 180 Angle Sum Theorem (x + 4) + (8x – 4) + 90 = 180Substitution 9x + 90= 180Simplify. 9x= 90Subtract 90 from each side. x= 10Divide each side by 9. Answer: So, m  B = 8(10) – 4 or 76.

27. Example 4 A.8 B.16 C.22 D.28 ALGEBRA Find m  D.

28. Concept

29. Example 5 Find Angle Measures INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find m  S and m  T.

30. Example 5 Find Angle Measures Since TSUV is inscribed in a circle, opposite angles are supplementary. m  S + m  V = 180 m  U + m  T = 180 m  S + 90 = 180(14x) + (8x + 4)= 180 m  S = 9022x + 4= 180 22x= 176 x= 8 Answer: So, m  S = 90 and m  T = 8(8) + 4 or 68.

31. Example 5 A.48 B.36 C.32 D.28 INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find m  N.

32. End of the Lesson