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Geometry

Geometry. 9.5 Inscribed Angles. Inscribed Angles. The vertex is on the circle The sides of the angle: Are chords of the circle Intercept an arc on the circle. Inscribed angle. Intercepted Arc. Inscribed Angle Theorem.

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Geometry

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  1. Geometry 9.5 Inscribed Angles

  2. Inscribed Angles • The vertex is on the circle • The sides of the angle: • Are chords of the circle • Intercept an arc on the circle Inscribed angle Intercepted Arc

  3. Inscribed Angle Theorem • The measure of the inscribed angle is half the measure of its central angle (and therefore half the intercepted arc). 30o 80o 160o 60o 60o 160o

  4. A Very Similar Theorem • The measure of the angle created by a chord and a tangent equals half the intercepted arc. tangent 50o tangent 35o chord chord 100o 70o

  5. Corollary • If two inscribed angles intercept the same arc, then the angles are congruent. ~ sf = giants ~ x = y y x giants sf

  6. Corollary • If an inscribed angle intercepts a semicircle, then it is a right angle. Why? 180o diameter diameter 90o

  7. Corollary • If a quadrilateral is inscribed in a circle, then opposite angles are supplementary. 70o 85o supplementary supplementary 95o 110o

  8. 140 60 y 20 110 O y 20 O O y x x x Solve for the variables. 140o 90o 100o 75o 20o 150o Semicircle 120o 40o Angle x and the 20o angle intercept the same arc. x = 40o x = 60o y = 75o x = 20o y = 50o y = 90o

  9. 80 x x y y O O O x y 82 z z Solve for the variables. x and y both intercept a semicircle. Inscribed Quadrilateral x = 98o Part of semicircle 100o supplementary 180o 100o y + 82o + z = 180o x = 40o y + z = 98o x = 90o The red and orange arcs are congruent (they have congruent chords). y = 50o y = 90o Thus, y and z are congruent angles (they intercept the red and orange arcs). z = 90o y = 49o z = 49o

  10. 4x B B A A 5x 50 x2 15x 8x C C D D x2 Find x and the measure of angle D. Inscribed Quadrilateral supplementary If x is negative, this angle would have a negative value. 100o If x is negative, this angle would have a negative value. X2 + 15x = 100 X2 + 8x = 180 X2 + 15x - 100 = 0 X2 + 8x - 180 = 0 ( )( ) = 0 x + x - 20 5 ( )( ) = 0 x + x - 18 10 x + 20 = 0 and x – 5 = 0 x + 18 = 0 and x – 10 = 0 x = -20 and x = 5 x = -18 and x = 10

  11. HW • P. 352-355 CE #1-9 WE #1-9, 19-21

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