Inscribed Angles Inscribed Angles An angle whose vertex

Inscribed Angles

Inscribed Angles • An angle whose vertex is ON the circle and whose sides are chords of the circle is an inscribed angle. • An arc whose endpoints are on the inscribed angle is an intercepted arc. Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.

Finding Measures of Arcs and Inscribed Angles • Find the measure of the blue arc.

Finding Measures of Arcs and Inscribed Angles • Find the measure of the blue arc. • 115°

Comparing Measures of Inscribed Angles • Find m ACB, m ADB, and m AEB. The measure of each angle is half the measure of • 60

Using the Inscribed Angle Theorem • What are the values of a and b?

What are m A, m B, m C, and m D?

Corollaries to the Inscribed Angle Theorem Corollary 1: Two inscribed angles that intercept the same arc are congruent. Corollary 2: An angle inscribed in a semicircle is a right angle. Corollary 3: The opposite angles of a quadrilateral inscribed in a circle are supplementary.

Using Corollaries to Find Angle Measures • What is the measure of each numbered angle?

What is the measure of each numbered angle?

• Find the value of each variable. • DEFG is inscribed in a circle, so opposite angles are supplementary. • m D + m F = 180° • z + 80 = 180 • z = 100 • z° • 120° • 80° • y°

• Find the value of each variable. • DEFG is inscribed in a circle, so opposite angles are supplementary. • m E + m G = 180° • y + 120 = 180 • y = 60 • z° • 120° • 80° • y°

Tangents and Intercepted Arcs Theorem 12 -12: The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

Using Arc Measure • In the diagram, SR is a tangent to the circle at Q. If m. PMQ = 212˚, what is m PQR?

If KJ is tangent to O, what are the values of x and y?