Angles of Circles Vocabulary Inscribed Angle An angle

Angles of Circles

Vocabulary • Inscribed Angle- An angle whose vertex is ON the circle and whose sides contain chords of the circle. • Intercepted Arc- The arc that lies in the interior of an inscribed angle and has endpoints on the angle.

• If two inscribed angles of a circle intercept the same arc, then the angles T are congruent. S R V • A polygon is an inscribed polygon if all of its vertices lie on a circle. The circle that contains these vertices is a circumscribed circle.

3 Possibilities for Angles 1. Angle lies INSIDE the circle. M L J K P 2. Angle lies OUTSIDE the circle. M K L 3. Angle lies ON the circle. L K M

Angle measures 1. The measure of an angle that lies INSIDE the circle: Add Arcs 2 M J K L P 2. The measure of an angle that lies OUTSIDE the circle: Subtract Arcs 2 M L J K P 3. The measure of an angle that lies ON the L circle: K M

• If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. If one side of an inscribed triangle is a diameter, then the triangle is a right triangle. • A quadrilateral can be inscribed in a circle only if it’s opposite angles are supplementary.

Example 1 Find the value of the indicated angle or arc. a. m<T b. KL T S R L V 23° K 16° M

Example 2 Find the indicated arc measures. a. m. ST R b. m. PR T • J 72° 15° c. m<PJQ P d. m. PQR S Q

Example 3 Find the measure of the indicated angle or arc in circle J. m<LMP = 54° a. b. c. d. e. f. g. m. LP m<KLM m<NLM m<LNP m<KLN m<MPN m. NK L K J P N 54° M 78°