Inscribed Angles 10-4. Using Inscribed Angles An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.

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Presentation transcript:

Inscribed Angles 10-4

Using Inscribed Angles An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle.

Measure of an Inscribed Angle If an angle is inscribed in a circle, then its measure is one half the measure of its intercepted arc. m  ADB = ½m

Ex. 1: Finding Measures of Arcs and Inscribed Angles Find the measure of the blue arc or angle. M<NMP = ½ 100°

Example A B C D 70 o 7x o Find x.

Theorem 9-5 If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then the angles are congruent.

Comparing Measures of Inscribed Angles

Ex. 3: Finding the Measure of an Angle It is given that m  E = 75 °. What is m  F? 75°

Theorem 9-6 If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle.

Example A B C D E m AED= ? (

Ex. 1: Finding Measures of Arcs and Inscribed Angles Find the measure of the blue arc or angle.

Find the value of each variable. 2x°

Ex: find x. P C Q R3x o

Theorem 9-7 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. D, E, F, and G lie on some circle, C, if and only if m  D + m  F = 180 ° and m  E + m  G = 180 °

Find the value of each variable. 120° 80° y°y° z°z°

That’s all folks! Class work Page 727 problems 1-6, 8-20 Homework Page 728 problems 28-30