1. Lesson 4 Menu 1.Refer to the figure. The radius of is 35, = 80, LM = 45, and LM  NO. Find. 2.Find. 3.Find the measure of NO. 4.Find the measure of NT. 5.Find the measure of RT.

2. Lesson 4 MI/Vocab intercepted Find measures of inscribed angles. Find measures of angles of inscribed polygons.

3. Lesson 4 TH1

4. Lesson 4 Ex1 Measures of Inscribed Angles

5. Lesson 4 Ex1 Measures of Inscribed Angles Arc Addition Postulate Simplify. Subtract 168 from each side. First determine Divide each side by 2.

6. Lesson 4 Ex1 Measures of Inscribed Angles So, m

7. Lesson 4 Ex1 Measures of Inscribed Angles

8. A.A B.B C.C D.D Lesson 4 CYP1 A.30 B.60 C.15 D.120

9. A.A B.B C.C D.D Lesson 4 CYP1 A.110 B.55 C.125 D.27.5

10. A.A B.B C.C D.D Lesson 4 CYP1 A.30 B.80 C.40 D.10

11. A.A B.B C.C D.D Lesson 4 CYP1 A.110 B.55 C.125 D.27.5

12. A.A B.B C.C D.D Lesson 4 CYP1 A.110 B.55 C.125 D.27.5

13. Lesson 4 TH2

14. Lesson 4 Ex2 Proof with Inscribed Angles Given: Prove: ΔPJK  ΔEHG

15. Lesson 4 Ex2 Proof with Inscribed Angles Proof: Statements Reasons 1. Given1. 2. 2. If 2 chords are, corr. minor arcs are. 4. 4. Inscribed angles of arcs are. 5. 5. Right angles are congruent. 6. ΔPJK  ΔEHG 6. AAS 3. 3. Definition of intercepted arc

16. Lesson 4 CYP2 Choose the best reason to complete the following proof. Given: Prove: ΔCEM  ΔHJM

17. Lesson 4 CYP2 1. Given 2. ______ 3. Vertical angles are congruent. 4. Radii of a circle are congruent. 5. ASA Proof: Statements Reasons 1. 2. 3. 4. 5. ΔCEM  ΔHJM

18. Lesson 4 CYP2 1.A 2.B 3.C 4.D A.Alternate Interior Angle Theorem B.Substitution C.Definition of  angles D.Inscribed angles of  arcs are .

19. Lesson 4 Ex3 Inscribed Arcs and Probability

20. Lesson 4 Ex3 Inscribed Arcs and Probability The probability that is the same as the probability of L being contained in.

21. 1.A 2.B 3.C 4.D Lesson 4 CYP3 A.B. C.D.

22. Lesson 4 TH3

23. Lesson 4 Ex4 Angles of an Inscribed Triangle

24. Lesson 4 Ex4 Angles of an Inscribed Triangle ΔUVT and ΔUVT are right triangles. m  1 = m  2 since they intercept congruent arcs. Then the third angles of the triangles are also congruent, so m  3 = m  4. Angle Sum Theorem Simplify. Subtract 105 from each side. Divide each side by 3.

25. Lesson 4 Ex4 Angles of an Inscribed Triangle Use the value of x to find the measures of Given Answer:

26. Lesson 4 Ex5 Draw a sketch of this situation. Angles of an Inscribed Quadrilateral

27. Lesson 4 Ex5 Angles of an Inscribed Quadrilateral To find we need to know To find first find Inscribed Angle Theorem Sum of arcs in circle = 360 Subtract 174 from each side.

28. Lesson 4 Ex5 Angles of an Inscribed Quadrilateral Inscribed Angle Theorem Substitution Divide each side by 2. Since we now know three angles of a quadrilateral, we can easily find the fourth. m  Q + m  R + m  S + m  T=360360° in a quadrilateral 87 + 102 + 93 + m  T=360Substitution m  T=78Subtraction Answer: m  S = 93; m  T = 78

29. Lesson 4 TH4