9.6. Inscribed Quadrilaterals in Circles http://www.ck12.org
This investigation shows that the opposite angles in an inscribed quadrilateral are supplementary. By cutting the
quadrilateral in half, through the diagonal, we were able to show that the other two angles (that we did not cut
through) formed a linear pair when matched up.
Inscribed Quadrilateral Theorem:A quadrilateral is inscribed in a circle if and only if the opposite angles are
supplementary.
Example A
Find the value of the missing variable.
x+ 80 ◦= 180 ◦by the Inscribed Quadrilateral Theorem.x= 100 ◦.
y+ 71 ◦= 180 ◦by the Inscribed Quadrilateral Theorem.y= 109 ◦.
Example B
Find the value of the missing variable.
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 ◦.Find the value of the missing variables.
Example C
Findxandyin the picture below.
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 ◦Watch this video for help with the Examples above.
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Click image to the left for more content.CK-12 Foundation: Chapter9InscribedQuadrilateralsinCirclesB
Vocabulary
Acircleis the set of all points that are the same distance away from a specific point, called thecenter. Aradiusis
the distance from the center to the circle. Achordis a line segment whose endpoints are on a circle. Adiameter
is a chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
Acentral angleis an angle formed by two radii and whose vertex is at the center of the circle. Aninscribed angle