Section 10-3 Inscribed Angles

Inscribed angles An angle whose vertex is on a circle and whose sides contain chords of the circle. A B D is an inscribed angle.

Intercepted arc The arc that lies in the interior of an inscribed angle and has endpoints on the angle.

Measure of an Inscribed Angle Theorem If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

A R T Example: If then m = If m = then =

An angle inscribed in a semicircle is a right angle. C A T Circle S S Q

Theorem 10-9 If two inscribed angles intercept the same arc, then the angles are congruent. 2 1

INSCRIBED Inside another shape CircumSCRIBED Outside another shape

If all the vertices of a polygon lie on the circle, the polygon is inscribed in the circle and the circle is circumscribed about the polygon.

When each side of a polygon is tangent to a circle, the polygon is said to be circumscribed about the circle and the circle is inscribed in the polygon.

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Theorem 10-10

D G O Therefore, is a diameter of the circle.

Theorem If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Q A U D are supplementary are supplementary