Geometry 9.5 Inscribed Angles

Inscribed Angles The vertex is on the circle The sides of the angle: AAre chords of the circle I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). 30 o 60 o 160 o 80 o 160 o

A Very Similar Theorem The measure of the angle created by a chord and a tangent equals half the intercepted arc. 50 o 100 o tangent chord 70 o t a n g e n t c h o r d 35 o

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

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

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

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

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

Find x and the measure of angle D. A B C D 5x 8x x2x2 A B C D 4x 15x 50  x2x Inscribed Quadrilateral supplementary X 2 + 8x = 180 X 2 + 8x = 0 ( )( ) = 0x + x x + 18 = 0 and x – 10 = 0 x = -18 and x = 10 If x is negative, this angle would have a negative value. 100 o X x = 100 X x = 0 ( )( ) = 0x + x x + 20 = 0 and x – 5 = 0 x = -20 and x = 5 If x is negative, this angle would have a negative value.

HW P CE #1-9 WE #1-9, 19-21