Theorem 12-9: The measure of an inscribed angles is half the measure of its intercepted arc. m  B= 1 / 2 mAC ( B A C

Example” Find the measure of  A A B C D m  A= 1 / 2 mBCD ( m  A= 1 / 2 ( ) m  A= 1 / 2 (150 0 ) m  A=75 0

Example” Find the measure of  D A B C D m  D= 1 / 2 mABC ( m  D= 1 / 2 ( ) m  D= 1 / 2 (190 0 ) m  D=95 0

Corollaries #1 Two inscribed angles that intercept the same arc are congruent. m  B  m  C B C

Corollaries #2 An angle inscribed in a semicircle is a right angle m  B=90 0 B

Corollaries #3 The opposite angles of a quadrilateral inscribed in a circle are supplementary. m  A+m  C=180 0 m  B+m  D =180 0 B D A C

Example” Find the measure of a and b. a b0b0 O 32 0 A is inscribed in a semi- circle,  a is a right angle

Example” Find the measure of a and b. a b0b0 O 32 0 a=90 0 The sum of the angles of a triangle is 180 0,  the other angle is =

Example” Find the measure of a and b. a b0b0 O 32 0 a= = 1 / 2 b   =b

Theorem 12-10: The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. m  C= 1 / 2 mBDC ( B D C

Example: RS and TU are diameters of  A. RB is tangent to  A at point R. Find m  BRT and m  TRS. B R U S A T

m  BRT B R U S A T m  BRT= 1 / 2 m RT ) mRT=mURT-mUR ) ) ) mRT= ) mRT=54 0 ) m  BRT= 1 / 2 (54 0 ) m  BRT=27 0

m  TRS B R U S A T m  BRS=mBRT+m  TRS =27 0 +m  TRS 63 0 =m  TRS