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Lesson 9.3 Arcs pp. 381-387

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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.

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

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.

L

K

M

LKM is a central angle.

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.

L

N

K

M

LNM is an inscribed angle.

A

B

C

Since mABC = 60°, then mAC = 60 also.

60

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

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

Definition

A semicircle is an arc measuring 180°.

Postulate 9.2

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

Theorem 9.8

Major Arc Theorem.mACB = 360 - mAB.

EXAMPLE If mAB = 50, find mACB.

mACB = 360 – mAB

mACB = 360 – 50

mACB = 310

Definition

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

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.

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.

Theorem 9.10

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

A

X

C

Z

B

Y

If B Y and ABC XYZ,

then AC XZ

Theorem 9.10

A

X

C

Z

B

Y

If B Y and AC XZ,

then ABC XYZ

Theorem 9.10

Theorem 9.11

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

A

X

C

Z

B

Y

If B Y and ABC XYZ,

then AC XZ

Theorem 9.11

A

X

C

Z

B

Y

If B Y and AC XZ,

then ABC XYZ

Theorem 9.11

Theorem 9.12

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

A

X

C

Z

B

Y

If B Y and ABC XYZ,

then AC XZ

Theorem 9.12

A

X

C

Z

B

Y

If B Y and AC XZ,

then ABC XYZ

Theorem 9.12

A

E

M

30°

45°

D

60°

B

C

Find mAB.

A

E

M

30°

45°

D

60°

B

C

Find mAE.

A

E

M

30°

45°

D

60°

B

C

Find mDC + mDE.

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

1.36

2.54

3.72

4.108

D

C

108

M

A

B

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

1.36

2.54

3.72

4.108

D

C

108

M

A

B

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

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

Homework

pp. 385-387

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.

►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

►A. Exercises

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

Find each of

the following.

7.mBOD

F

B

50

O

C

A

30

40

E

10

G

D

= 90

►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)

►A. Exercises

Use the figure for exercises 11-13.

C

D

P

Q

A

B

11.If AB CD and mBPA = 80, find

mCQD.

mCQD= 80 (Thm. 9.10)

►A. Exercises

Use the figure for exercises 11-13.

C

D

P

Q

A

B

13.If mBPA = 75 and mCQD = 75, what is true about AB and CD? Why?

Given:mAB + mACB = m☉P

Prove:mACB = 360 - mAB

C

P

A

B

►B. Exercises

Prove the following theorems.

14.Theorem 9.8

►B. Exercises

Prove the following theorems.

15.Given:☉U with XY YZ ZX

Prove: ∆XYZ is an equilateral triangle

X

Y

U

Z

►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

►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

■ Cumulative Review

24.State the Triangle Inequality.

■ Cumulative Review

25.State the Exterior Angle Inequality.

■ Cumulative Review

26.State the Hinge Theorem.

■ Cumulative Review

27.State the greater than property.

■ Cumulative Review

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