10-4 Inscribed Angles The student will be able to:

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Presentation transcript:

10-4 Inscribed Angles The student will be able to: Find measures of inscribed angles. Find measures of angles of inscribed polygons.

Inscribed Angles Inscribed Angles – have a vertex on a circle and sides that contain chords of the circle. In C, is an inscribed angle. Intercepted arc – has endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle. In C, minor arc is intercepted by . There are three ways that an angle can be inscribed in a circle.

If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc. and Example 1: Find each measure:

Related Inscribed Angles If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. and both intercept So, Example 2: If and , find = 3(8) = 24°

Angles of Inscribed Polygons An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. If is a semicircle, then If then is a semicircle and is a diameter. Example 3: If and , find x. ΔFGH is a right triangle because inscribes a semicircle. 7x + 2 +17x – 8 + 90 = 180 24x + 84 = 180 24x = 96 x = 4

Quadrilaterals Inscribed in a Circle Only certain quadrilaterals can be inscribed in a circle. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. If quadrilateral KLMN is inscribed in A, then and are supplementary and and are supplementary. Example 4: Quadrilateral WXYZ is inscribed in V. Find and x + 95 = 180 x + 60 = 180 x = 85 x = 120

You Try It: 1. Find 2. An insignia is and emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find and x + 4 + 8x – 4 + 90 = 180 = 8(10) - 4 9x = 90 = 80 - 4 x = 10 = 76 x + 90 = 180 14x + 8x+4 = 180 x = 90 22x = 176 x = 8 = 8(8) + 4 = 68