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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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Lesson 8 5 l.jpg
Lesson 8-5

Angle Formulas

Lesson 8-5: Angle Formulas


Central angle l.jpg
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)

Lesson 8-5: Angle Formulas


Central angle theorem l.jpg

Y

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


Slide4 l.jpg

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 l.jpg
Inscribed Angle

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 l.jpg
Intercepted Arc

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


Slide7 l.jpg

Inscribed Angle Theorem

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


Slide8 l.jpg

A

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


Slide9 l.jpg

An angle inscribed in a semicircle is a right angle.

P

180

90

S

R

Lesson 8-5: Angle Formulas


Interior angle theorem l.jpg

A

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


Slide11 l.jpg

Example: Interior Angle Theorem

91

A

C

B

D

85

Lesson 8-5: Angle Formulas


Exterior angles l.jpg

°

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


Exterior angle theorem l.jpg
Exterior Angle Theorem

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

Lesson 8-5: Angle Formulas


Example exterior angle theorem l.jpg
Example: Exterior Angle Theorem

Lesson 8-5: Angle Formulas


Slide15 l.jpg

D

6

C

E

Q

5

3

A

F

2

1

4

G

30°

25°

100°

Lesson 8-5: Angle Formulas


Slide16 l.jpg

Inscribed Quadrilaterals

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


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