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10 - 4 Inscribed Angles

10 - 4 Inscribed Angles. Inscribed Angle: An angle that has its vertex on the circle and its sides contained in chords of the circle. Vertex B is on the circle. B. Arc ADC is the arc intercepted by angle ABC. AB and BC are chords of the circle. A. C. Theorem 10.5 Inscribed Angle Theorem.

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10 - 4 Inscribed Angles

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  1. 10 - 4 Inscribed Angles

  2. Inscribed Angle: An angle that has its vertex on the circle and its sides contained in chords of the circle Vertex B is on the circle B Arc ADC is the arc intercepted by angle ABC. AB and BC are chords of the circle A C

  3. Theorem 10.5 Inscribed Angle Theorem • If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle.

  4. Example 1 p. 579 • Use circle O on pg 579 & review the example shown. mAB = 140,mBC=100, mAD=mDC. Find the measures of angle 4 & 5. • mAB = 140 therefore, measure of angle 4 is ½ of 140. Measure of angle 4 = 70 • mBC = 100 therefore, measure of angle 5 is ½ of 100. Measure of angle 5 = 50

  5. Theorem 10.6: If two inscribed angles of a circle (or congruent circles) intercept congruent arcs or the same arc, then the angles are congruent. B B A C A D F C D E

  6. Review the Proof in Ex. 2 p. 580 • Try check your progress #2 • Answer: Statements (Reasons) • 1) RT bisects SU (Given) • 2) SV = VU (def of segment bisector) • 3) Angle SRT intercepts arc ST. Angle SUT intercepts arc ST. (def of intercepted arc) • 4) Angle SRT = angle SUT (inscribed angles of same arc are congruent) • 5) Angle RVS = angle UVT (vertical angles are congruent • 6) Triangle RVS = Triangle UVT (AAS)

  7. Theorem 10.7 • If the inscribed angle of a triangle intercepts a semicircle, the angle is a right angle. A D Arc ADC is a semicircle, so the measure of angle ABC is 90. B C

  8. Refer to Circle F & given info in Ex. 4 on pg.581 • Find the measure of angle 3 & angle 4. • Answer • Since Arc AD = Arc BD, then angle 3 & angle 4 are also equal. Therefore, each are 45 since Arc ADE is a semicircle so angle B is 90 leaving the other two angles (3 & 4) are complementary (add to 90).

  9. Refer to Circle V on pg. 582 • Quadrilateral WXYZ is inscribed in circle V. If the measure of angle W = 95 and measure of angle Z is 60, find the measure of angle X and Y. • Arc WXY = 120 & Arc ZYX = 190 • This means that arc WZY = 360-120 = 240 & XWZ = 360-190 = 170 • Angle Y is ½ of 170 = 85 • Angle X is ½ of 240 = 120

  10. Homework #67 Study Guide 10.4 Worksheet

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