By: Justin Mitchell and Daniel Harrast. Inscribed angle- an angle whose vertex is on a circle and whose sides contain chords of the circle. Intercepted.

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Presentation transcript:

By: Justin Mitchell and Daniel Harrast

Inscribed angle- an angle whose vertex is on a 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. Intercepted arc Inscribed angle

The measure of an inscribed angle is one half the measure of its intercepted arc. CD A B m<ADB=1/2mAB

Ex. 1: Use inscribed angles  Find the indicated measure of m<T in P. Solution m<T= ½ mRS = ½ (48˚) = 24˚ Q R S T P 48˚

Find the indicated measure of m<A A B C 84˚ 42˚

Find mRS and < STR. What do you notice about <STR and <RUS? Solution From theorem 10.7, you know that mRS = 2m<RUS = 2(31˚) = 62˚. Also, m<STR = ½ mRS = ½ (62˚) = 31˚. So, <STR is congruent to <RUS. 31˚ S T U R

 Find the indicated measure of mRS R Q S 67˚ 134˚

If two inscribed angles of a circle intercept the same arc, then the angles are congruent. A D B C <ADB is congruent to <ACB

Inscribed polygon: A polygon whose vertices all lie on a circle. Circumscribed circle: A circle that contains the vertices of the polygon. Circumscribed circle Inscribed polygon

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is a right angle. A B C Dm< ABC = 90˚. If and only if AC is a diameter of a circle.

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

PQRS is inscribed in a circle, so opposite angles are supplementary. m<P + m<R = 180˚ 75˚+ y˚ = 180˚ y = 105˚ m<Q + m<S = 180˚ 80˚ + x˚ = 180˚ x = 100˚ Q R P S 80˚ y˚ 75˚ x˚