Section 9.5 Inscribed Angles 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 vertex at the center of a circle. angle equals arc B A If AB = 100° then AOB = 100°

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 O B C A AB is the intercepted arc of inscribed angle ACB

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

Example 1 Find x, y, z x° y° 90° x= ½ (80°) = 40° y= 2(55°) = 110°

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

Example 3 Find BC  1 = 40°, BC= 2(40°) = 80° B C

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

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

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

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

Example 7 Find x, y, z O is the center 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°

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

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°

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

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

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

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

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

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