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Circles

Circles. Section 9.5 Inscribed Angles. Warm Up. Circles Crossword puzzle Some answers are 2 words, do not leave blanks. Homework Answers p. 347. 1. 8 2. 5 3. 9√2 4. 55 5. 80 6. 45 7. 24 8. 12 9. 10√5 . Central Angle.

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Circles

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  1. Circles Section 9.5 Inscribed Angles

  2. Warm Up • Circles Crossword puzzle • Some answers are 2 words, do not leave blanks

  3. Homework Answers p. 347 1. 8 2. 5 3. 9√2 4. 55 5. 80 6. 45 7. 24 8. 12 9. 10√5

  4. Central Angle vertex at the center of a circle. angle equals arc A If AB = 100° then AOB = 100° O B

  5. Inscribed Angle An angle whose vertex is on the edge of a circle. C is a point on circle O ACB is an inscribed angle A C O AB is the intercepted arc of inscribed angle ACB B

  6. Theorem 9.7 The measure of an inscribed angle is equal to HALF the intercepted arc A If AB = 100° then ACB = 50° C O B

  7. Example 1 Find x, y, z x° y° 90° x= ½ (80°) = 40° y= 2(55°) = 110° z= 360°-(90°+80°+110°)= 80° 80° 55° z°

  8. Example 2 Find 1 • BC = 70°,  1 = ½ (70°) = 35° B C 2. BC = 84°,  1 = ½ (84°) = 42° 3. BAC = 280°, BC = 360°- 280°= 80°  1 = ½ (80°) = 40° 1 A

  9.  1 = 40°, BC= 2(40°) = 80° Example 3 Find BC B C 2.  1 = 36° ,BC = 2(36°) = 72° 1 A

  10. Theorem 9.7 – Corollary 1If two inscribed angles intersect the same arc, then the angles are congruent. B C 1 intersects arc BC 2 intersects arc BC 1 = 2 2 1 A

  11. Theorem 9.7 – Corollary 2An angle inscribed in a semicircle is a right angle. B O is the center of the circle arc AB is a semicircle 1 intersects arc AB O 1 R A 1 = ½ (AB) = 90°

  12. Theorem 9.7 – Corollary 3If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. B 3 O is the center of the circle BXAZ is a quadrilateral inscribed in circle O O 2 X • 1 and  2 are opposite angles • 1 +  2 = 180° 1 Z 4 A

  13. Example 6 Find x, y, z 160° x= ½ (160°) = 80° a°+ 80° + 68° = 180° a° = 32° z = 2(32°) = 64° a° Y intersects arcs 160° + 64° y= ½(160° + 64°) = 112° y° 68° 80° x° z° 64°

  14. Example 7 Find x, y, zO is the center x° x and y are inscribed in the same semicircle x = y = ½ (180°) = 90° 2 chords are marked congruent, therefore the 2 arcs are congruent. The 2 arcs are inscribed in a semicircle Therefore z = ½ (180°) = 90° O y° z° 68°

  15. Example 8 Find x, y, zO is the center AB is a semicircle = 180° CB = 180°- 110° = 70° y is a cental angle = arc y = 70° C 110° x° y° A 43° x = ½ (CB) = ½(70°) = 35° O B z°= 2(43°) = 86° z°

  16. Example 9 Find x, y, z A quadrilateral is inscribed in the circle 60° 110° x° and 85° are opposite angles Therefore they are supplementary x= 180°- 85° = 95° z° 85° y° and 110° are opposite angles Therefore they are supplementary y= 180°- 110° = 70° x° y intersects arcs z° + 60° y= ½(60° + z°) 70°= ½ (60° + z°) 140° = (60° + z°) z° = 80° 70° y°

  17. Theorem 9.8The measure of an angle formed by a chord and a tangent is equal to HALF its intercepted arc A vertex on the edge of the circle G If CB = 120° then FCB = 60° ACB = _______ GCB= _______ C O F B

  18. Examples 10-12 CBD = ½(240°) = 120° A B D ABD = ½(120°) = 60° BD = 360°- 240° = 120° C O 240°

  19. Examples 13-15 PRT = ½(100°) = 50° PRQ = ½(170°) = 85° 170° Q QRS = ½(QR) QR = 360°- (170°+100°) = 90° QRS = ½(90°) = 45° s O P R 100° T

  20. Examples 16-18AC is tangent to circle Z at B BE is a diameter, therefore ABC =CBE= 90° A F EBD = 90°- 75° = 15° DE = 2(15°) = 30° DB = 2(75°) = 150° B E Z 75° D C

  21. Examples 19-21AC is tangent to circle Z at B BED = ½ (150°) = 75° BDE is inscribed in a semicircle BDE = 90° A F B E 15° Z 75° BFE is a semicircle BFE = 180° 30° D C 150°

  22. Cool Down • Complete exercises 1-6 on bottom of notesheet • Check your answers with Mrs. Baumher • Begin HW p. 354 1-9, 19-21

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