10 - 4 Inscribed Angles.

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Presentation transcript:

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 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.

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

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 A B C A D F D C E

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)

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

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).

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

Homework #66 Study Guide 10.4 Worksheet