More Angle-Arc Theorems

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

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 X A B Y • Theorem 89: • If two inscribed or tangent-chord angles intercept the same arc, then they are congruent

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

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

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

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

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

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

10. A Special Theorem About Tangent-Tangent Angles T O P S 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.

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

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

13. Problem 1 - Proof V S E O L N s ʘO 1. Given V S 2. If two inscribed s interceptthe same arc, they are . 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

14. Problem 2 A C B O 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.

15. Problem 2 - Solution A C B O Since A is inscribed in a semicircle, it is a right angle. By the Pythagorean Theorem, (AB)2 + (AC)2 = (BC)2 (AB)2 + 402 = 412 AB = 9 mm

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

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

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

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

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

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

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

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

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