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lesson 8-5 - PowerPoint PPT Presentation

Lesson 8-5. Angle Formulas. Central Angle. Definition:. An angle whose vertex lies on the center of the circle. NOT A Central Angle (of a circle). Central Angle (of a circle). Central Angle (of a circle). Y. 110 . 110 . O. Z. Central Angle Theorem.

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Angle Formulas

Lesson 8-5: Angle Formulas

Definition:

An angle whose vertex lies on the center of the circle.

NOT A Central Angle

(of a circle)

Central Angle

(of a circle)

Central Angle

(of a circle)

Lesson 8-5: Angle Formulas

110

110

O

Z

Central Angle Theorem

The measure of a center angle is equal to the measure of the intercepted arc.

Center Angle

Intercepted Arc

Example:

Give is the diameter, find the value of x and y and z in the figure.

Lesson 8-5: Angle Formulas

Example: Find the measure of each arc.

4x + 3x + (3x +10) + 2x + (2x-14) = 360°

14x – 4 = 360°

14x = 364°

x = 26°

4x = 4(26) = 104°

3x = 3(26) = 78°

3x +10 = 3(26) +10= 88°

2x = 2(26) = 52°

2x – 14 = 2(26) – 14 = 38°

Lesson 8-5: Angle Formulas

Inscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle).

Examples:

3

1

2

4

Yes!

No!

No!

Yes!

Lesson 8-5: Angle Formulas

Intercepted Arc: An angle intercepts an arc if and only if each of the following conditions holds:

1. The endpoints of the arc lie on the angle.

2. All points of the arc, except the endpoints, are in the interior of the angle.

3. Each side of the angle contains an endpoint of the arc.

Lesson 8-5: Angle Formulas

The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.

Y

Inscribed Angle

110

55

Z

Intercepted Arc

An angle formed by a chord and a tangent can be considered an inscribed angle.

Lesson 8-5: Angle Formulas

F

A

°

°

40

y

D

°

50

B

°

°

y

50

B

°

x

C

°

C

x

E

E

Examples: Find the value of x and y in the fig.

Lesson 8-5: Angle Formulas

P

180

90

S

R

Lesson 8-5: Angle Formulas

D

1

B

C

Interior Angle Theorem

Definition:

Angles that are formed by two intersecting chords.

2

E

Interior Angle Theorem:

The measure of the angle formed by the two intersecting chords is equal to ½ the sum of the measures of the intercepted arcs.

Lesson 8-5: Angle Formulas

91

A

C

B

D

85

Lesson 8-5: Angle Formulas

1

y

°

2

y

°

°

x

3

x

°

y

°

x

Exterior Angles

An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle.

Two secants

2 tangents

A secant and a tangent

Lesson 8-5: Angle Formulas

The measure of the angle formed is equal to ½ the difference of the intercepted arcs.

Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas

6

C

E

Q

5

3

A

F

2

1

4

G

30°

25°

100°

Lesson 8-5: Angle Formulas

If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary.

mDAB + mDCB = 180 

mADC + mABC = 180 

Lesson 8-5: Angle Formulas