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Warm Up Section 4.5 Find x : 1. 2. 3.

Warm Up Section 4.5 Find x : 1. 2. 3. 4. 5. 6. x o. 32 o. x o. 70 o. x o. x. 12. 45 o. 100 o. x o. x o. Answers to Warm Up Section 4.5 Find x :

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Warm Up Section 4.5 Find x : 1. 2. 3.

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  1. Warm Up Section 4.5 Find x: 1. 2. 3. 4. 5. 6. xo 32o xo 70o xo x 12 45o 100o xo xo

  2. Answers to Warm Up Section 4.5 Find x: 1. 2. 3. 4. 5. 6. 32o 140o 90o xo 32o xo 70o xo x 12 45o 100o xo xo 40o 45o

  3. Angles Formed byChords, Secants, and Tangents Section 4.5 Standard: MM2G3 bd Essential Question: How are properties of chords, tangents, and secants used to find angle measures?

  4. Recall how arcs are related to central angles: arc = angle Ex. x = 65

  5. Recall how an arc is related to its inscribed angle: angle = ½ arc arc = angle × 2 angle = arc ÷ 2 Ex . Ex . x = 80 x = 32

  6. Y The measure of the angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. X 1 Z W angle = ½ (arc 1 + arc 2)

  7. Example 1: If = 45o and = 75o, find m m1 = ½ (mRS + mPQ) m1 = ½(75o + 45o) m1 = ½(120o) m1 = 60o R Q 75o 45o 1 P S

  8. Example 2: If and 80o , find m1 = ½ (mRS + mPQ) 55o = ½(80o + x) 110o = (80o + x) 30o = x R Q 80o x 55o P S

  9. Try these with your partner: 3. 4. x° 80° x° 100° 20° xo = ½(80o + 20o) xo = ½(100o) xo = 50o 90o = ½(100o + xo) 180o = (100o+ xo) 80o = xo

  10. 70° 5. yo x° 40° yo = ½(70o + 40o) yo= ½(110o) yo = 55o xo = 180o – 55o = 125o

  11. Case 2: Vertex outside the circle.The measure of an angle formed by 2 secants, 2 tangents, or a secant and a tangent is half the difference of the measures of the intercepted arcs. angle = ½ (arc 1 – arc 2)

  12. Example 6: If mDC = 100o and mEB = 40o, find mA. D E A B C 100o 40o xo mA = ½ (mDC – mEB) xo= ½(100o – 40o) xo= ½ (60o) xo= 30o

  13. Example 7: If mW = 65o and mXZ = 70o, find mXVY Y Z W 70o 65o xo V mW = ½ (mXVY – mXZ) 65o = ½(xo– 70o) 130o = xo– 70o 200o = xo X

  14. Example 8: If mQRS = 240o, find mQS and mP. S mQS = 360o – 240o = 120o R 120o 240o xo P mP = ½ (mQRS – mQS) xo= ½(240o – 120o) xo= ½ (120o) xo= 60o Q

  15. Try these with your partner: 9. 10. x° 140° 230° 20o 80° x° 120° 130o xo = ½(230o – 130o) xo = ½ (100o) xo = 50o xo = ½(80o – 20o) xo = ½ (60o) xo = 30o

  16. 11. 12. 35° x° 30° x° 150° 50o 100° 35o = ½(150o – xo) 70o = (150o – xo) -80o = – xo 80o = xo xo = ½(50o – 30o) xo = ½ (20o) xo = 10o

  17. Summary: Measures of Angles Formed by Radii, Chords, Tangents and Secants x° 1 The vertex of the angle is located at the center of the circle. So, the angle is a central angle and is equal to the measure of the intercepted arc. m1 = xo angle = arc

  18. The vertex of the angle is a point on the circle. So, the measure of the angle is one half the measure of the intercepted arc. angle = ½arc m2 = ½ xo x° x° 2 2

  19. The vertex of the angle is located in the interior of the circle and not at the center, so the measure of the angle is half the sum of the intercepted arcs. angle = ½(arc1 + arc2) m3 = ½(xo + yo) y° 3 x°

  20. x° The vertex of the angle is located in the exterior of the circle and not at the center, so the measure of the angle is half the difference of the intercepted arcs. angle = ½(arc1 – arc2) m4 = ½(xo –yo) x° x° y° y° y° 4 4 4

  21. BE is a diameter of the circle with center O. AT is tangent to the circle at A. mAB = 80o, mBC = 20o, and mDE = 50o. 100o 10 E A 9 4 1 6 5 50o 3 O 2 80o D 8 7 B 20o C 110o

  22. 100o 10 E A 9 4 1 6 5 50o 3 O 2 80o D 8 7 B 20o C 110o 13. m1 = ½(80o) = 40o 14. m2 = ½(100o) = 50o 15. m3 = ½(80o+ 50o) = 65o 16. m4 = ½(100o– 50o) = 25o 17. m5 = ½(80o) = 40o 18. m6 = ½(180o) = 90o

  23. 100o 10 E A 9 4 1 6 5 50o 3 O 2 80o D 8 7 B 20o C 110o 19. m7 = ½(100o) = 50o 20. m8 = 20o 21. m9 = ½(50o) = 25o 22. m10 = ½(150o) = 75o

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