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Lesson 9.3 Arcs pp. 381-387 PowerPoint PPT Presentation


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Lesson 9.3 Arcs pp. 381-387. Objectives: 1.To identify and define relationships between arcs of circles, central angles, and inscribed angles. 2.To identify minor arcs, major arcs, and semicircles and express them using correct notation.

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Lesson 9.3 Arcs pp. 381-387

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Lesson 9 3 arcs pp 381 387

Lesson 9.3

Arcs

pp. 381-387


Lesson 9 3 arcs pp 381 387

Objectives:

1.To identify and define relationships between arcs of circles, central angles, and inscribed angles.

2.To identify minor arcs, major arcs, and semicircles and express them using correct notation.

3.To prove theorems relating the measure of arcs, central angles, and chords.


Lesson 9 3 arcs pp 381 387

Definition

A central angle is an angle that is in the same plane as the circle and whose vertex is the center of the circle.


Lesson 9 3 arcs pp 381 387

L

K

M

LKM is a central angle.


Lesson 9 3 arcs pp 381 387

Definition

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

Arc measure is the same measure as the degree measure of the central angle that intercepts the arc.


Lesson 9 3 arcs pp 381 387

L

N

K

M

LNM is an inscribed angle.


Lesson 9 3 arcs pp 381 387

A

B

C

Since mABC = 60°, then mAC = 60 also.

60


Lesson 9 3 arcs pp 381 387

A minor arc is an arc measuring less than 180. Minor arcs are denoted with two letters, such as AB, where A and B are the endpoints of the arc.

Definition


Lesson 9 3 arcs pp 381 387

A major arc is an arc measuring more than 180. Major arcs are denoted with three letters, such as ABC, where A and C are the endpoints and B is another point on the arc.

Definition


Lesson 9 3 arcs pp 381 387

Definition

A semicircle is an arc measuring 180°.


Lesson 9 3 arcs pp 381 387

Postulate 9.2

Arc Addition Postulate. If B is a point on AC, then mAB + mBC = mAC.


Lesson 9 3 arcs pp 381 387

Theorem 9.8

Major Arc Theorem.mACB = 360 - mAB.


Lesson 9 3 arcs pp 381 387

EXAMPLE If mAB = 50, find mACB.

mACB = 360 – mAB

mACB = 360 – 50

mACB = 310


Lesson 9 3 arcs pp 381 387

Definition

Congruent Arcs are arcs on congruent circles that have the same measure.


Lesson 9 3 arcs pp 381 387

A

X

C

Z

B

Y

If B  Y and AC  XZ, then AC  XZ

Theorem 9.9

Chords on congruent circles are congruent if and only if they subtend congruent arcs.


Lesson 9 3 arcs pp 381 387

A

X

C

Z

B

Y

If B  Y and AC  XZ, then AC  XZ

Theorem 9.9

Chords on congruent circles are congruent if and only if they subtend congruent arcs.


Lesson 9 3 arcs pp 381 387

Theorem 9.10

In congruent circles, chords are congruent if and only if the corresponding central angles are congruent.


Lesson 9 3 arcs pp 381 387

A

X

C

Z

B

Y

If B  Y and ABC  XYZ,

then AC  XZ

Theorem 9.10


Lesson 9 3 arcs pp 381 387

A

X

C

Z

B

Y

If B  Y and AC  XZ,

then ABC  XYZ

Theorem 9.10


Lesson 9 3 arcs pp 381 387

Theorem 9.11

In congruent circles, minor arcs are congruent if and only if their corresponding central angles are congruent.


Lesson 9 3 arcs pp 381 387

A

X

C

Z

B

Y

If B  Y and ABC  XYZ,

then AC  XZ

Theorem 9.11


Lesson 9 3 arcs pp 381 387

A

X

C

Z

B

Y

If B  Y and AC  XZ,

then ABC  XYZ

Theorem 9.11


Lesson 9 3 arcs pp 381 387

Theorem 9.12

In congruent circles, two minor arcs are congruent if and only if the corresponding major arcs are congruent.


Lesson 9 3 arcs pp 381 387

A

X

C

Z

B

Y

If B  Y and ABC  XYZ,

then AC  XZ

Theorem 9.12


Lesson 9 3 arcs pp 381 387

A

X

C

Z

B

Y

If B  Y and AC  XZ,

then ABC  XYZ

Theorem 9.12


Lesson 9 3 arcs pp 381 387

A

E

M

30°

45°

D

60°

B

C

Find mAB.


Lesson 9 3 arcs pp 381 387

A

E

M

30°

45°

D

60°

B

C

Find mAE.


Lesson 9 3 arcs pp 381 387

A

E

M

30°

45°

D

60°

B

C

Find mDC + mDE.


Lesson 9 3 arcs pp 381 387

Given circle M with diameters DB and AC, mAD = 108. Find mAMB.

1.36

2.54

3.72

4.108

D

C

108

M

A

B


Lesson 9 3 arcs pp 381 387

Given circle M with diameters DB and AC, mAD = 108. Find mBMC.

1.36

2.54

3.72

4.108

D

C

108

M

A

B


Lesson 9 3 arcs pp 381 387

Given circle M with diameters DB and AC, mAD = 108. Find mDAB.

1.90

2.180

3.360

4.Don’t know

D

C

108

M

A

B


Lesson 9 3 arcs pp 381 387

Given circle M with diameters DB and AC, mAD = 108. Find mDC.

1.36

2.54

3.72

4.108

D

C

108

M

A

B


Lesson 9 3 arcs pp 381 387

Homework

pp. 385-387


Lesson 9 3 arcs pp 381 387

F

B

50

O

C

A

30

40

E

10

G

D

►A. Exercises

Use the diagram for exercises 1-10. In circle O, AC is a diameter.


Lesson 9 3 arcs pp 381 387

►A. Exercises

Use the diagram for exercises 1-10. In circle O, AC is a diameter.

Find each of

the following.

5.mAB

F

B

50

O

C

A

30

40

E

10

G

D

= 130


Lesson 9 3 arcs pp 381 387

►A. Exercises

Use the diagram for exercises 1-10. In circle O, AC is a diameter.

Find each of

the following.

7.mBOD

F

B

50

O

C

A

30

40

E

10

G

D

= 90


Lesson 9 3 arcs pp 381 387

►A. Exercises

Use the diagram for exercises 1-10. In circle O, AC is a diameter.

Find each of

the following.

9.mBC + mBA

F

B

50

O

C

A

30

40

E

10

G

D

= 180 (Post. 9.2)


Lesson 9 3 arcs pp 381 387

►A. Exercises

Use the figure for exercises 11-13.

C

D

P

Q

A

B

11.If AB  CD and mBPA = 80, find

mCQD.

mCQD= 80 (Thm. 9.10)


Lesson 9 3 arcs pp 381 387

►A. Exercises

Use the figure for exercises 11-13.

C

D

P

Q

A

B

13.If mBPA = 75 and mCQD = 75, what is true about AB and CD? Why?


Lesson 9 3 arcs pp 381 387

Given:mAB + mACB = m☉P

Prove:mACB = 360 - mAB

C

P

A

B

►B. Exercises

Prove the following theorems.

14.Theorem 9.8


Lesson 9 3 arcs pp 381 387

►B. Exercises

Prove the following theorems.

15.Given:☉U with XY  YZ  ZX

Prove: ∆XYZ is an equilateral triangle

X

Y

U

Z


Lesson 9 3 arcs pp 381 387

►B. Exercises

Prove the following theorems.

16.Given: Points M, N, O, and P on ☉L;

MO  NP

Prove: MP  NO

P

M

O

L

N


Lesson 9 3 arcs pp 381 387

►B. Exercises

Prove the following theorems.

17.Given:☉O; E is the midpoint of BD

and AC; BE  AE

Prove: MP  NO

B

A

E

O

C

D


Lesson 9 3 arcs pp 381 387

■ Cumulative Review

24.State the Triangle Inequality.


Lesson 9 3 arcs pp 381 387

■ Cumulative Review

25.State the Exterior Angle Inequality.


Lesson 9 3 arcs pp 381 387

■ Cumulative Review

26.State the Hinge Theorem.


Lesson 9 3 arcs pp 381 387

■ Cumulative Review

27.State the greater than property.


Lesson 9 3 arcs pp 381 387

■ Cumulative Review

28.Prove that the surface area of a cone is always greater than its lateral surface area.


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