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## PowerPoint Slideshow about ' Lesson 9.3 Arcs pp. 381-387' - ashton

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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.

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

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.

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

A semicircle is an arc measuring 180°.

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

Major Arc Theorem.mACB = 360 - mAB.

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

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.

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.

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

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

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

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

pp. 385-387

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

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

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)

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)

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?

Prove:mACB = 360 - mAB

C

P

A

B

►B. Exercises

Prove the following theorems.

14. Theorem 9.8

Prove the following theorems.

15. Given:☉U with XY YZ ZX

Prove: ∆XYZ is an equilateral triangle

X

Y

U

Z

Prove the following theorems.

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

MO NP

Prove: MP NO

P

M

O

L

N

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

24. State the Triangle Inequality.

25. State the Exterior Angle Inequality.

26. State the Hinge Theorem.

27. State the greater than property.

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

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