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10.3 Inscribed Angles

This guide explores the concepts of inscribed angles and intercepted arcs in circles. An inscribed angle is formed by points on the circle, with its vertex on the circle and sides that are chords. The relationship between an inscribed angle and the intercepted arc is highlighted by key theorems. Notably, the measure of the inscribed angle is half that of the intercepted arc. Additionally, it discusses essential properties such as congruency of inscribed angles intercepting the same arc and the conditions for inscribing quadrilaterals in circles.

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10.3 Inscribed Angles

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  1. 10.3 Inscribed Angles

  2. Definitions • 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 (arc AB)

  3. Theorems • If an angle is inscribed in a circle, then its measure is half the measure of the intercepted arc. • m < APB = ½ AB • m < APB = 60

  4. Theorems • If two inscribed angles of a circle intercept the same arc, then the angles are congruent. • m<DCR = m<RAD

  5. Theorems 3. If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.

  6. Last Theorem!!!! 4. A quadrilateral can be inscribed in a circle if and only if its opposite sides are supplementary.

  7. Examples • Find the measure of arc TR • 260 • Find the m<DIS • 21

  8. Last Example 3. Find the measure of angle D and angle W

  9. Homework • P. 617 9-23all

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