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 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

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

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

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

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

Solve for the variables. 1. 2. 3. 140o O y 20 110 x O y 20 x 60 140 O y x 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

Solve for the variables. x and y both intercept a semicircle. 4. 5. 6. Inscribed Quadrilateral O x 80 y O x y z O x z y 82 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

Find x and the measure of angle D. Inscribed Quadrilateral A B C D 4x 15x 50 x2 7. 8. A B C D 5x 8x x2 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

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