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10.5 Apply Other Angle Relationships in Circles Hubarth Geometry Theorem 10.11 If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. C . B 2 1 A m1 1 mAB 2 m2 1 mBCA 2 Ex 1 Find Angle and Arc Measures Line m is tangent to the circle. Find the measure of the red angle or arc. a. m 1 = 1 2 (130o) = 65o b. m KJL = 2(125o) = 250o Theorem 10.12 Angles Inside the Circle Theorem D If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measure of the arcs intercepted by the angle and its vertical angle. A 1 2 B C m1 1 (mDC mAB) 2 m2 1 (mAB mBC ) 2 Theorem 10.13 Angles Outside the Circle Theorem If a tangent and a secant, two tangents or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. P X B A Q W 3 1 2 Z Y R C 1 1 1 m 3 (mXY mWZ ) m2 (mPQR mPR) m1 (mBC mAC ) 2 2 2 . Ex 2 Find an Angle Measure Inside a Circle Find the value of x. xo = x o = 1 (mJM + mLK) 2 1 o o (130 + 156 ) 2 xo = 143 Ex 3 Find and Angle Measure Outside a Circle Find the value of x. The tangent CD and the secant CB intersect outside the circle. m 1 BCD = 2 (mAD – mBD) xo = x 1 o o (178 – 76 ) 2 = 51 Practice Find the indicated measure. m 1= 1 2 (210o) = 105o m RST = 2(98o) = 196o m XY = o 2(80o) = 160 Find the value of the variable. 5. o y = 61 6. a = 104o xo 253.7o