1 / 24

# More Angle-Arc Theorems

More Angle-Arc Theorems. Section 10.6. A. T. X. P. A. O. O. B. C. A. B. B. P. C. Y. S. D. Objectives. Recognize congruent inscribed and tangent-chord angles Determine the measure of an angle inscribed in a semicircle

Télécharger la présentation

## More Angle-Arc Theorems

An Image/Link below is provided (as is) to download presentation Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

### Presentation Transcript

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

More Related