1. More Angle-Arc Theorems Section 10.6 X Y A B P C A B D O B A C O T P S

2. Objectives Recognize congruent inscribed and tangent-chord angles Determine the measure of an angle inscribed in a semicircle Apply the relationship between the measures of a tangent-tangent angle and its minor arc

3. InCongruent Inscribed and Tangent-Chord Angles – Theorem 89: If two inscribed or tangent-chord angles intercept the same arc, then they are congruent X Y A B

4. InCongruent Inscribed and Tangent-Chord Angles Given: X and Y are inscribed angles intercepting arc AB Conclusion: X Y X Y A B

5. Congruent Inscribed and tangent Chord Angles – Theorem 90: If two inscribed or tangent-chord angles intercept congruent arcs, then they are congruent P C A B D E

6. Congruent Inscribed and tangent Chord Angles If is the tangent at D and AB CD, we may conclude that P CDE. P C A B D E

7. Angles Inscribed in Semicircles – Theorem 91: An angle inscribed in a semi circle is a right angle O B A C

8. A Special Theorem About Tangent-Tangent Angles – Theorem 92: The sum of the measure of a tangent and its minor arc is 180 o O T P S

9. A Special Theorem About Tangent-Tangent Angles Given: and are tangent to circle O. Prove: m P + mTS = 180 O T P S

10. A Special Theorem About Tangent-Tangent Angles Proof: Since the sum of the measures of the angles in quadrilateral SOTP is 360 and since T and S are right angles, m P + m O = 180. Therefore, m P + mTS = 180. O T P S

11. Sample Problems X Y A B P C A B D O B A C O T P S

12. Problem 1 Given: ʘ O Conclusion: ∆LVE ∆NSE, EV ● EN = EL ● SE V S N L O E

13. Problem 1 - Proof 1. ʘ O 1. Given 2. V S 2. If two inscribed s intercept the same arc, they are. 3. L N 3. Same as 2 4. ∆LVE ∆NSE 4. AA (2,3) 5. = 5. Ratios of corresponding sides of ~ ∆ are =. 6. EV ● EN = EL ● SE 6. Means-Extremes Products Theorem V S N L O E s

14. Problem 2 In Circle O, is a diameter and the radius of the circle is 20.5 mm. Chord has a length of 40 mm. Find AB. O A C B

15. Problem 2 - Solution Since A is inscribed in a semicircle, it is a right angle. By the Pythagorean Theorem, (AB) 2 + (AC) 2 = (BC) 2 (AB) 2 + 40 2 = 41 2 AB = 9 mm O A C B

16. Problem 3 Given: ʘ O with tangent at B, ‖ Prove: C BDC A B N D O C

17. Problem 3 - Proof 1. is tangent to ʘ O. 1. Given 2. ‖ 2. Given 3. ABD BDC 3. ‖ lines ⇒ alt. int. s 4. C ABD4. If an inscribed and a tangent-chord intercept the same arc, they are. 5. C BDC5. Transitive Property A B N D O C

18. You try!! X Y A B P C A B D O B A C O T P S

19. Problem A Given: and are tangent segments. QR = 163 o Find: a. P b. PQR Q P R

20. Solution a. 163 o + P = 180 o P = 17 o b.PQR PRQ (intercept the same arc) 17 o + 2x = 180 o 2x = 163 o x = 81.5 o PQR = 81.5 o Q P R

21. Problem B Given: A, B, and C are points of contact. AB = 145 o, Y = 48 o Find: Z B X C Y Z A

22. Solution X + 145 o = 180 o X = 35 o X + Y + Z = 180 o 35 o + 48 o + Z = 180 o Z = 97 o B X C Y Z A

23. Problem C In the figure shown, find m P. A P B (6x) o (15x + 33) o

24. Solution P + AB = 180 o 6x + 15x + 33 o = 180 o 21x = 147 o x = 7 o m P = 6x m P = 42 o A P B (6x) o (15x + 33) o