Section 5-8Properties of Circles

Wed, Feb 02

Essential Questions• What are the relationships among parts of

a circle?

• What are the properties of circles and how do you apply them?

• Where you’ll see this:

• Market research, food service, art, recreation, navigation

Wed, Feb 02

Vocabulary1. Circle:

2. Radius:

3. Chord:

4. Diameter:

5. Central Angle:

Wed, Feb 02

Vocabulary1. Circle: All points that are the same distance from a

fixed center point; 360° total2. Radius:

3. Chord:

4. Diameter:

5. Central Angle:

Wed, Feb 02

Vocabulary1. Circle: All points that are the same distance from a

fixed center point; 360° total2. Radius: A segment whose endpoints are the center

of a circle and on the circle3. Chord:

4. Diameter:

5. Central Angle:

Wed, Feb 02

Vocabulary1. Circle: All points that are the same distance from a

fixed center point; 360° total2. Radius: A segment whose endpoints are the center

of a circle and on the circle3. Chord: A segment where both endpoints are on the

circle4. Diameter:

5. Central Angle:

Wed, Feb 02

Vocabulary1. Circle: All points that are the same distance from a

fixed center point; 360° total2. Radius: A segment whose endpoints are the center

of a circle and on the circle3. Chord: A segment where both endpoints are on the

circle4. Diameter: A chord that goes through the center of a

circle5. Central Angle:

Wed, Feb 02

Vocabulary1. Circle: All points that are the same distance from a

fixed center point; 360° total2. Radius: A segment whose endpoints are the center

of a circle and on the circle3. Chord: A segment where both endpoints are on the

circle4. Diameter: A chord that goes through the center of a

circle5. Central Angle: An angle where the vertex is the

center of the circle

Wed, Feb 02

Vocabulary6. Arc:

7. Semicircle:

8. Minor Arc:

9. Major Arc:

10. Inscribed Angle:

Wed, Feb 02

Vocabulary6. Arc: A section of the circumference of a circle7. Semicircle:

8. Minor Arc:

9. Major Arc:

10. Inscribed Angle:

Wed, Feb 02

Vocabulary6. Arc: A section of the circumference of a circle7. Semicircle: An arc that is half of the circumference;

half a circle8. Minor Arc:

9. Major Arc:

10. Inscribed Angle:

Wed, Feb 02

Vocabulary6. Arc: A section of the circumference of a circle7. Semicircle: An arc that is half of the circumference;

half a circle8. Minor Arc: An arc that is less than half the

circumference; same measure as the central angle9. Major Arc:

10. Inscribed Angle:

Wed, Feb 02

Vocabulary6. Arc: A section of the circumference of a circle7. Semicircle: An arc that is half of the circumference;

half a circle8. Minor Arc: An arc that is less than half the

circumference; same measure as the central angle9. Major Arc: An arc that is more than half the

circumference10. Inscribed Angle:

Wed, Feb 02

Vocabulary6. Arc: A section of the circumference of a circle7. Semicircle: An arc that is half of the circumference;

half a circle8. Minor Arc: An arc that is less than half the

circumference; same measure as the central angle9. Major Arc: An arc that is more than half the

circumference10. Inscribed Angle: An angle whose vertex is on the

circle and whose sides are chords of the circle; half the measure of the arc it contains

Wed, Feb 02

Circle

Wed, Feb 02

Radius

Wed, Feb 02

Chord

Wed, Feb 02

Diameter

Wed, Feb 02

Central Angle

Wed, Feb 02

Arc

Wed, Feb 02

Semicircle

Wed, Feb 02

Minor Arc

Wed, Feb 02

Major Arc

Wed, Feb 02

Inscribed Angle

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132°

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

x°

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

x° x°

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

x° x°

x + x +132 + 82 = 360

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

x° x°

x + x +132 + 82 = 360

2x + 214 = 360

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

x° x°

x + x +132 + 82 = 360

2x + 214 = 360 −214 −214

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

x° x°

x + x +132 + 82 = 360

2x + 214 = 360 −214 −214

2x =146

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

x° x°

x + x +132 + 82 = 360

2x + 214 = 360 −214 −214

2x =146 2 2

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

x° x°

x + x +132 + 82 = 360

2x + 214 = 360 −214 −214

2x =146 2 2 x = 73

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

x + x +132 + 82 = 360

2x + 214 = 360 −214 −214

2x =146 2 2 x = 73

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠ABC =

12

(mAD + mCD )

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠ABC =

12

(mAD + mCD )

=

12

(73+ 73)

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠ABC =

12

(mAD + mCD )

=

12

(73+ 73) =

12

(146)

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠ABC =

12

(mAD + mCD )

=

12

(73+ 73) =

12

(146) = 73°

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠BCD =

12

(mAD + mAB )

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠BCD =

12

(mAD + mAB )

=

12

(73+132)

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠BCD =

12

(mAD + mAB )

=

12

(73+132) =

12

(205)

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠BCD =

12

(mAD + mAB )

=

12

(73+132) =

12

(205) =102.5°

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠CDA =

12

(mBC + mAB )

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠CDA =

12

(mBC + mAB )

=

12

(82 +132)

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠CDA =

12

(mBC + mAB )

=

12

(82 +132) =

12

(214)

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠CDA =

12

(mBC + mAB )

=

12

(82 +132) =

12

(214) =107°

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠DAB =

12

(mBC + mCD )

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠DAB =

12

(mBC + mCD )

=

12

(82 + 73)

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠DAB =

12

(mBC + mCD )

=

12

(82 + 73) =

12

(155)

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠DAB =

12

(mBC + mCD )

=

12

(82 + 73) =

12

(155) = 77.5°

Wed, Feb 02

Example 1In circle O, AD ≅ CD . Find the measures of the

angles of quadrilateral ABCD, when

mAB =132° and mBC = 82°.

132° 82°

73° 73°

m∠ABC = 73°

m∠BCD =102.5°

m∠CDA =107°

m∠DAB = 77.5°

Wed, Feb 02

Example 2Identify the following for circle P.

a. Diameter b. Radius

c. Chord

h. Central Angle

d. mLM

e. mLMK f. mLJ

g. m∠LKJ

Wed, Feb 02

Example 2Identify the following for circle P.

a. Diameter b. Radius

c. Chord

h. Central Angle

d. mLM

e. mLMK f. mLJ

g. m∠LKJ

JK

Wed, Feb 02

Example 2Identify the following for circle P.

a. Diameter b. Radius

c. Chord

h. Central Angle

d. mLM

e. mLMK f. mLJ

g. m∠LKJ

JK KP

Wed, Feb 02

Example 2Identify the following for circle P.

a. Diameter b. Radius

c. Chord

h. Central Angle

d. mLM

e. mLMK f. mLJ

g. m∠LKJ

JK KP

KL

Wed, Feb 02

Example 2Identify the following for circle P.

a. Diameter b. Radius

c. Chord

h. Central Angle

d. mLM

e. mLMK f. mLJ

g. m∠LKJ

JK KP

KL = 62° + 47°

Wed, Feb 02

Example 2Identify the following for circle P.

a. Diameter b. Radius

c. Chord

h. Central Angle

d. mLM

e. mLMK f. mLJ

g. m∠LKJ

JK KP

KL = 62° + 47° =109°

Wed, Feb 02

Example 2Identify the following for circle P.

a. Diameter b. Radius

c. Chord

h. Central Angle

d. mLM

e. mLMK f. mLJ

g. m∠LKJ

JK KP

KL = 62° + 47° =109°

= 62° +180°

Wed, Feb 02

Example 2Identify the following for circle P.

a. Diameter b. Radius

c. Chord

h. Central Angle

d. mLM

e. mLMK f. mLJ

g. m∠LKJ

JK KP

KL = 62° + 47° =109°

= 62° +180° = 242°

Wed, Feb 02

Example 2Identify the following for circle P.

a. Diameter b. Radius

c. Chord

h. Central Angle

d. mLM

e. mLMK f. mLJ

g. m∠LKJ

JK KP

KL = 62° + 47° =109°

= 62° +180° = 242° = 62°

Wed, Feb 02

Example 2Identify the following for circle P.

a. Diameter b. Radius

c. Chord

h. Central Angle

d. mLM

e. mLMK f. mLJ

g. m∠LKJ

JK KP

KL = 62° + 47° =109°

= 62° +180° = 242° = 62°

= 1

2(62°)

Wed, Feb 02

Example 2Identify the following for circle P.

a. Diameter b. Radius

c. Chord

h. Central Angle

d. mLM

e. mLMK f. mLJ

g. m∠LKJ

JK KP

KL = 62° + 47° =109°

= 62° +180° = 242° = 62°

= 1

2(62°) = 31°

Wed, Feb 02

Example 2Identify the following for circle P.

a. Diameter b. Radius

c. Chord

h. Central Angle

d. mLM

e. mLMK f. mLJ

g. m∠LKJ

JK KP

KL = 62° + 47° =109°

= 62° +180° = 242° = 62°

= 1

2(62°) = 31° ∠JPM

Wed, Feb 02

Problem Set

Wed, Feb 02

Problem Set

p. 228 #1-25 odd

“We are so accustomed to disguise ourselves to others that in the end we become disguised to ourselves.”

- Francois de La RochefoucauldWed, Feb 02