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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

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## More Angle-Arc Theorems

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**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 Apply the relationship between the measures of a tangent-tangent angle and its minor arc**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**InCongruent Inscribed and Tangent-Chord Angles**X A B Y Given: X and Y are inscribed angles intercepting arc AB Conclusion: X Y**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**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.**Angles Inscribed in Semicircles**O B A C • Theorem 91: • An angle inscribed in a semi circle is a right angle**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**A Special Theorem About Tangent-Tangent Angles**T O P S Given: and are tangent to circle O. Prove: m P + mTS = 180**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.**Sample Problems**A T X P A O O B C A B B P C Y S D**Problem 1**V S E O L N Given: ʘO Conclusion: ∆LVE ∆NSE, EV ● EN = EL ● SE**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**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.**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**Problem 3**B A C D O N Given: ʘO with tangent at B, ‖ Prove: C BDC**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**You try!!**A T X P A O O B C A B B P C Y S D**Problem A**Q P R Given: and are tangent segments. QR = 163o Find: a. P b. PQR**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**Problem B**X A B Y Z C Given: A, B, and C are points of contact. AB = 145o, Y = 48o Find: Z**Solution**X A B Y Z C X + 145o = 180o X = 35o X + Y + Z = 180o 35o + 48o + Z = 180o Z = 97o**Problem C**A (6x)o (15x + 33)o P B In the figure shown, find m P.**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

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