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 circleB

AB and BC are chords of the circle

Arc ADC is the arc intercepted by angle ABC.

AC

Theorem 10.5 Inscribed Angle Theorem

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

Example 1 p. 579

Use circle O on pg 579 & review the example Use circle O on pg 579 & review the example shown. mAB = 140,mBC=100, mAD=mDC. shown. mAB = 140,mBC=100, mAD=mDC. Find the measures of angle 4 & 5.Find the measures of angle 4 & 5.

mAB = 140 therefore, measure of angle 4 is ½ mAB = 140 therefore, measure of angle 4 is ½ of 140. Measure of angle 4 = 70of 140. Measure of angle 4 = 70

mBC = 100 therefore, measure of angle 5 is ½ mBC = 100 therefore, measure of angle 5 is ½ of 100. Measure of angle 5 = 50of 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.

A

D C

B

A

F

E

D

C

B

Review the Proof in Ex. 2 p. 580Try check your progress #2Try check your progress #2

Answer: Statements (Reasons)Answer: Statements (Reasons)

1) RT bisects SU (Given)1) RT bisects SU (Given)

2) SV = VU (def of segment bisector)2) SV = VU (def of segment bisector)

3) Angle SRT intercepts arc ST. Angle SUT intercepts 3) Angle SRT intercepts arc ST. Angle SUT intercepts arc ST. (def of intercepted arc)arc ST. (def of intercepted arc)

4) Angle SRT = angle SUT (inscribed angles of same 4) Angle SRT = angle SUT (inscribed angles of same arc are congruent)arc are congruent)

5) Angle RVS = angle UVT (vertical angles are 5) Angle RVS = angle UVT (vertical angles are congruentcongruent

6) Triangle RVS = Triangle UVT (AAS6) Triangle RVS = Triangle UVT (AAS))

Theorem 10.7

If the inscribed angle of a triangle intercepts a If the inscribed angle of a triangle intercepts a semicircle, the angle is a right angle.semicircle, the angle is a right angle.

A

B C

DArc ADC is a semicircle, so the measure of angle ABC is 90.

Refer to Circle F & given info in Ex. 4 on pg.581

Find the measure of angle 3 & angle 4.Find the measure of angle 3 & angle 4.

AnswerAnswer

Since Arc AD = Arc BD, then angle 3 & angle 4 Since Arc AD = Arc BD, then angle 3 & angle 4 are also equal. Therefore, each are 45 since are also equal. Therefore, each are 45 since Arc ADE is a semicircle so angle B is 90 Arc ADE is a semicircle so angle B is 90 leaving the other two angles (3 & 4) are leaving the other two angles (3 & 4) are complementary (add to 90).complementary (add to 90).

Refer to Circle V on pg. 582

Quadrilateral WXYZ is inscribed in circle V. If the Quadrilateral WXYZ is inscribed in circle V. If the measure of angle W = 95 and measure of angle Z measure of angle W = 95 and measure of angle Z is 60, find the measure of angle X and Y.is 60, find the measure of angle X and Y.

Arc WXY = 120 & Arc ZYX = 190Arc WXY = 120 & Arc ZYX = 190

This means that arc WZY = 360-120 = 240 & XWZ This means that arc WZY = 360-120 = 240 & XWZ = 360-190 = 170= 360-190 = 170

Angle Y is ½ of 170 = 85Angle Y is ½ of 170 = 85

Angle X is ½ of 240 = 120Angle X is ½ of 240 = 120

Study Guide 10.4 WorksheetHomework #66