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

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

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

4. Over Lesson 10–3 A.A B.B C.C D.D 5-Minute Check 5 26.8 to the nearest tenth.

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

6. Then/Now Find measures of inscribed angles. Find measures of angles of inscribed polygons. In this lesson we will:

7. Vocabulary inscribed angle—An angle whose vertex lies on a circle and whose sides contain chords of the circle. intercepted arc—The arc formed by an inscribed angle.

8. Concept

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

11. Example 1 Use Inscribed Angles to Find Measures B. = 2(252) or 104

12. A.A B.B C.C D.D Example 1 47 A. Find m  C.

13. A.A B.B C.C D.D Example 1 96 B.

14. Concept

15. 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.

16. A.A B.B C.C D.D Example 2 49 ALGEBRA Find m  I.

17. 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.

18. 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.

19. 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.

20. 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.

21. A.A B.B C.C D.D Example 3 AAS Congruence Theorem

22. Concept

23. 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.

24. A.A B.B C.C D.D Example 4 22 ALGEBRA Find m  D.

25. Concept

26. 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.

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

28. A.A B.B C.C D.D Example 5 48 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.