http://www.ck12.org Chapter 9. Circles
Theorem 9-10:A quadrilateral is inscribed in a circle if and only if the opposite angles are supplementary.
Example 5:Find the value of the missing variables.
a)
b)
Solution:
a)x+ 80 ◦= 180 ◦by Theorem 9-10.x= 100 ◦.
y+ 71 ◦= 180 ◦by Theorem 9-10.y= 109 ◦.
b) It is easiest to figure outzfirst. It is supplementary with 93◦, soz= 87 ◦. Second, we can findx. xis an
inscribed angle that intercepts the arc 58◦+ 106 ◦= 164 ◦. Therefore, by the Inscribed Angle Theorem,x= 82 ◦.yis
supplementary withx, soy= 98 ◦.
Example 6:Algebra ConnectionFindxandyin the picture below.
Solution:The opposite angles are supplementary. Set up an equation forxandy.
( 7 x+ 1 )◦+ 105 ◦= 180 ◦ ( 4 y+ 14 )◦+( 7 y+ 1 )◦= 180 ◦
7 x+ 106 ◦= 180 ◦ 11 y+ 15 ◦= 180 ◦
7 x= 84 ◦ 11 y= 165 ◦
x= 12 ◦ y= 15 ◦Example 7:Findxandyin the picture below.