Lesson 9.3Arcs

pp. 381-387

Lesson 9.3Arcs

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.

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.

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

DefinitionDefinitionDefinitionDefinition

KK

LL

MM

LKM is a central angle.LKM is a central angle.

An An inscribed angleinscribed angle is an angle is an angle with its vertex on a circle and with its vertex on a circle and with sides containing chords with sides containing chords of the circle.of the circle.Arc measureArc measure is the same is the same measure as the degree measure as the degree measure of the central angle measure of the central angle that intercepts the arc.that intercepts the arc.

DefinitionDefinitionDefinitionDefinition

KK

LL

MM

LNM is an inscribed angle.LNM is an inscribed angle.

NN

BB

AA

CC

6060

Since mABC = 60°, then mAC = 60 also.Since mABC = 60°, then mAC = 60 also.

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

DefinitionDefinitionDefinitionDefinition

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

DefinitionDefinitionDefinitionDefinition

A A semicirclesemicircle is an arc is an arc measuring 180°. measuring 180°.

DefinitionDefinitionDefinitionDefinition

Postulate 9.2Arc Addition Postulate. If B is a point on AC, then mAB + mBC = mAC.

Postulate 9.2Arc Addition Postulate. If B is a point on AC, then mAB + mBC = mAC.

Theorem 9.8Major Arc Theorem. mACB = 360 - mAB.

Theorem 9.8Major Arc Theorem. mACB = 360 - mAB.

EXAMPLE If mAB = 50, find mACB.EXAMPLE If mAB = 50, find mACB.

mACB = 360 – mAB mACB = 360 – mAB

mACB = 360 – 50mACB = 360 – 50

mACB = 310mACB = 310

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

DefinitionDefinitionDefinitionDefinition

Theorem 9.9

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

Theorem 9.9

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

XX

YYZZ

AA

BB CC

If B Y and AC XZ, then AC XZIf 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 9.9

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

XX

YYZZ

AA

BB CC

If B Y and AC XZ, then AC XZIf B Y and AC XZ, then AC XZ

Theorem 9.10

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

Theorem 9.10

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

XX

YYZZ

AA

BB CC

If B Y and ABC XYZ, then AC XZ

If B Y and ABC XYZ, then AC XZ

Theorem 9.10Theorem 9.10

XX

YYZZ

AA

BB CC

If B Y and AC XZ, then ABC XYZ

If B Y and AC XZ, then ABC XYZ

Theorem 9.10Theorem 9.10

Theorem 9.11

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

Theorem 9.11

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

XX

YYZZ

AA

BB CC

If B Y and ABC XYZ, then AC XZ

If B Y and ABC XYZ, then AC XZ

Theorem 9.11Theorem 9.11

XX

YYZZ

AA

BB CC

If B Y and AC XZ, then ABC XYZ

If B Y and AC XZ, then ABC XYZ

Theorem 9.11Theorem 9.11

Theorem 9.12

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

Theorem 9.12

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

XX

YYZZ

AA

BB CC

If B Y and ABC XYZ, then AC XZ

If B Y and ABC XYZ, then AC XZ

Theorem 9.12Theorem 9.12

XX

YYZZ

AA

BB CC

If B Y and AC XZ, then ABC XYZ

If B Y and AC XZ, then ABC XYZ

Theorem 9.12Theorem 9.12

Find mAB.Find mAB.

AA

BB CC

DD

EEMM

30°30°45°45°

60°60°

Find mAE.Find mAE.

AA

BB CC

DD

EEMM

30°30°45°45°

60°60°

Find mDC + mDE.Find mDC + mDE.

AA

BB CC

DD

EEMM

30°30°45°45°

60°60°

Given circle M with diameters

DB and AC, mAD = 108. Find

mAMB.

1. 36

2. 54

3. 72

4. 108

Given circle M with diameters

DB and AC, mAD = 108. Find

mAMB.

1. 36

2. 54

3. 72

4. 108 AABB

CCDD

MM108108

AABB

CCDD

MM108108

Given circle M with diameters

DB and AC, mAD = 108. Find

mBMC.

1. 36

2. 54

3. 72

4. 108

Given circle M with diameters

DB and AC, mAD = 108. Find

mBMC.

1. 36

2. 54

3. 72

4. 108

AABB

CCDD

MM108108

Given circle M with diameters

DB and AC, mAD = 108. Find

mDAB.

1. 90

2. 180

3. 360

4. Don’t know

Given circle M with diameters

DB and AC, mAD = 108. Find

mDAB.

1. 90

2. 180

3. 360

4. Don’t know

AABB

CCDD

MM108108

Given circle M with diameters

DB and AC, mAD = 108. Find

mDC.

1. 36

2. 54

3. 72

4. 108

Given circle M with diameters

DB and AC, mAD = 108. Find

mDC.

1. 36

2. 54

3. 72

4. 108

Homeworkpp. 385-387Homeworkpp. 385-387

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

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

AA

EEGG DD

CC

BBFF

OO 5050

40403030

1010

AA

EEGG DD

CC

BBFF

OO 5050

40403030

1010

= 130= 130

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

Find each of the following.5. mAB

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

Find each of the following.5. mAB

AA

EEGG DD

CC

BBFF

OO 5050

40403030

1010

= 90= 90

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

Find each of the following.7. mBOD

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

Find each of the following.7. mBOD

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

Find each of the following.9. mBC + mBA

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

Find each of the following.9. mBC + mBA AA

EEGG DD

CC

BBFF

OO 5050

40403030

1010= 180 (Post. 9.2)= 180 (Post. 9.2)

►A. ExercisesUse the figure for exercises 11-13. ►A. ExercisesUse the figure for exercises 11-13.

AB

PQ

D

C

11. If AB CD and mBPA = 80, find mCQD.

11. If AB CD and mBPA = 80, find mCQD. mCQD = 80 (Thm. 9.10)mCQD = 80 (Thm. 9.10)

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

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

►A. ExercisesUse the figure for exercises 11-13. ►A. ExercisesUse the figure for exercises 11-13.

AB

PQ

D

C

►B. ExercisesProve the following theorems. 14. Theorem 9.8

►B. ExercisesProve the following theorems. 14. Theorem 9.8

Given: mAB + mACB = m☉PProve: mACB = 360 - mAB Given: mAB + mACB = m☉PProve: mACB = 360 - mAB CC

PP

AABB

►B. ExercisesProve the following theorems. 15. Given: ☉U with XY YZ ZX

Prove: ∆XYZ is an equilateral triangle

►B. ExercisesProve the following theorems. 15. Given: ☉U with XY YZ ZX

Prove: ∆XYZ is an equilateral triangle

XX YY

ZZ

UU

►B. ExercisesProve the following theorems. 16. Given: Points M, N, O, and P on ☉L;

MO NPProve: MP NO

►B. ExercisesProve the following theorems. 16. Given: Points M, N, O, and P on ☉L;

MO NPProve: MP NO MM PP

NN

LLOO

►B. ExercisesProve the following theorems. 17. Given: ☉O; E is the midpoint of BD

and AC; BE AEProve: MP NO

►B. ExercisesProve the following theorems. 17. Given: ☉O; E is the midpoint of BD

and AC; BE AEProve: MP NO AA BB

DD

OO CC

EE

■ Cumulative Review24. State the Triangle Inequality.■ Cumulative Review24. State the Triangle Inequality.

■ Cumulative Review25. State the Exterior Angle Inequality.■ Cumulative Review25. State the Exterior Angle Inequality.

■ Cumulative Review26. State the Hinge Theorem.■ Cumulative Review26. State the Hinge Theorem.

■ Cumulative Review27. State the greater than property.■ Cumulative Review27. State the greater than property.

■ Cumulative Review28. Prove that the surface area of a cone

is always greater than its lateral surface area.

■ Cumulative Review28. Prove that the surface area of a cone

is always greater than its lateral surface area.