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Angles, Arcs, and Chords

Angles, Arcs, and Chords. Advanced Geometry Circles Lesson 2. In a circle or in congruent circles, two minor arcs are congruent if their corresponding chords are congruent. Inscribed & Circumscribed. inside. surrounding. X is circumscribed about quadrilateral ABCD.

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Angles, Arcs, and Chords

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  1. Angles, Arcs, and Chords Advanced Geometry Circles Lesson 2

  2. In a circle or in congruent circles, two minor arcs are congruent if their corresponding chords are congruent.

  3. Inscribed & Circumscribed inside surrounding X is circumscribed about quadrilateral ABCD. Quadrilateral ABCD is inscribed in X. ALL vertices of the polygon must lie on the circle.

  4. Example: A circle is circumscribed about a regular pentagon. What is the measure of the arc between each pair of consecutive vertices?

  5. In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc.

  6. Example: Circle R has a radius of 16 centimeters. Radius is perpendicular to chord , which is 22 cm long. If m = 110, find m . Find RS.

  7. Example: Circle W has a radius of 10 centimeters. Radius is perpendicular to chord , which is 16 cm long. If m = 53, find m . Find JL.

  8. In a circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Example: Chords and are equidistant from the center. If the radius of P is 15 and EF = 24, find PR and RH.

  9. Inscribed Angles & Intercepted Arcs If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc.

  10. Example: In O, m = 140, m = 100. Find m 1, m 2, m 3, m 4, and m 5.

  11. If two inscribed angles of a circle intercept congruent arcs or the same arc, then the angles are congruent.

  12. Example: Find m∠2 if m∠2 = 5x – 6 and m∠1 = 3x + 18.

  13. If the inscribed angle intercepts a semicircle, then the angle is a right angle. Example: Triangles TVU and TSU are inscribed in P with . Find the measure of each numbered angle if m∠2 = x + 9 and m∠4 = 2x + 6.

  14. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Example: Quadrilateral QRST is inscribed in Q = 87 M. If m and m R = 102, find m S and m T.

  15. Probability Example: Points M and N are on a circle so that m = 80. Suppose point L is randomly located on the same circle so that it does not coincide with M or N. What is the probability that m MLN = 40?

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