Lesson 11-3 Lesson 11-3 Inscribed AnglesInscribed Angles

An angle is an inscribed angle if its vertex is on a circle and its sides are chords of the circle.

Definitions

B inscribed ais a .n ngle

For the circle above:

AC intercepted arc is the of B.

A

B

C

Theorem 11-9 Inscribed Angle Theorem

1B = AC

2m m

The measure of an inscribed angle is half the measure of its intercepted arc.

A

B

C

Find the values of the variables.Find the values of the variables.

Corollaries to the Inscribed Angle Theorem

1. Two inscribed angles that intercept the same arc are congruent.

A

B

C

D

D B

Corollaries to the Inscribed Angle Theorem

2. An angle inscribed in a semicircle is a right angle.

A

B

C

A

B

C

A

B

C

oIn each case B = . 90,

Corollaries to the Inscribed Angle Theorem

3. The opposite angles of a quadrilateral inscribed in a circle are supplementary.

A

B

C

D

oA + C = 180 . oB + D = 180 .

Find the values of the variables.Find the values of the variables.

Theorem 11-10

1B = ACB

2m m

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

A

B

C

B

A

C

Find the values of the variables.Find the values of the variables.

Yes; each is formed by a tangent and a chord, and they intercept the same arc.

45

m A = 100; m B = 75; m C = 80; m D = 105

m X = 80; m Y = 70; m Z = 90; m W = 120

No; the diagonal would be a diameter of O and the inscribed angle would be a right angle, which was not found in Exercise 1 above.

.

GEOMETRY LESSON 11-3GEOMETRY LESSON 11-3

In the diagram below, O circumscribes quadrilateral ABCD and is inscribed in quadrilateral XYZW.

1. Find the measure of each inscribed angle.

2. Find m DCZ.

3. Are XAB and XBA congruent? Explain.

4. Find the angle measures in quadrilateral XYZW.

.

5. Does a diagonal of quadrilateral ABCD intersect the center of the circle? Explain how you can tell.

Homework: Homework:

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