Lesson 9.3 Arcs pp. 381-387
description
Transcript of Lesson 9.3 Arcs pp. 381-387
![Page 1: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/1.jpg)
Lesson 9.3Arcs
pp. 381-387
Lesson 9.3Arcs
pp. 381-387
![Page 2: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/2.jpg)
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.
![Page 3: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/3.jpg)
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
![Page 4: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/4.jpg)
KK
LL
MM
LKM is a central angle.LKM is a central angle.
![Page 5: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/5.jpg)
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
![Page 6: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/6.jpg)
KK
LL
MM
LNM is an inscribed angle.LNM is an inscribed angle.
NN
![Page 7: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/7.jpg)
BB
AA
CC
6060
Since mABC = 60°, then mAC = 60 also.Since mABC = 60°, then mAC = 60 also.
![Page 8: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/8.jpg)
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
![Page 9: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/9.jpg)
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
![Page 10: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/10.jpg)
A A semicirclesemicircle is an arc is an arc measuring 180°. measuring 180°.
DefinitionDefinitionDefinitionDefinition
![Page 11: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/11.jpg)
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.
![Page 12: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/12.jpg)
Theorem 9.8Major Arc Theorem. mACB = 360 - mAB.
Theorem 9.8Major Arc Theorem. mACB = 360 - mAB.
![Page 13: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/13.jpg)
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
![Page 14: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/14.jpg)
Congruent ArcsCongruent Arcs are arcs on are arcs on congruent circles that have the congruent circles that have the same measure. same measure.
DefinitionDefinitionDefinitionDefinition
![Page 15: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/15.jpg)
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
![Page 16: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/16.jpg)
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
![Page 17: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/17.jpg)
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.
![Page 18: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/18.jpg)
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
![Page 19: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/19.jpg)
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
![Page 20: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/20.jpg)
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.
![Page 21: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/21.jpg)
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
![Page 22: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/22.jpg)
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
![Page 23: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/23.jpg)
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.
![Page 24: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/24.jpg)
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
![Page 25: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/25.jpg)
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
![Page 26: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/26.jpg)
Find mAB.Find mAB.
AA
BB CC
DD
EEMM
30°30°45°45°
60°60°
![Page 27: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/27.jpg)
Find mAE.Find mAE.
AA
BB CC
DD
EEMM
30°30°45°45°
60°60°
![Page 28: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/28.jpg)
Find mDC + mDE.Find mDC + mDE.
AA
BB CC
DD
EEMM
30°30°45°45°
60°60°
![Page 29: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/29.jpg)
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
![Page 30: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/30.jpg)
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
![Page 31: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/31.jpg)
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
![Page 32: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/32.jpg)
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
![Page 33: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/33.jpg)
Homeworkpp. 385-387Homeworkpp. 385-387
![Page 34: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/34.jpg)
►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
![Page 35: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/35.jpg)
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
![Page 36: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/36.jpg)
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
![Page 37: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/37.jpg)
►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)
![Page 38: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/38.jpg)
►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)
![Page 39: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/39.jpg)
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
![Page 40: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/40.jpg)
►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
![Page 41: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/41.jpg)
►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
![Page 42: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/42.jpg)
►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
![Page 43: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/43.jpg)
►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
![Page 44: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/44.jpg)
■ Cumulative Review24. State the Triangle Inequality.■ Cumulative Review24. State the Triangle Inequality.
![Page 45: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/45.jpg)
■ Cumulative Review25. State the Exterior Angle Inequality.■ Cumulative Review25. State the Exterior Angle Inequality.
![Page 46: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/46.jpg)
■ Cumulative Review26. State the Hinge Theorem.■ Cumulative Review26. State the Hinge Theorem.
![Page 47: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/47.jpg)
■ Cumulative Review27. State the greater than property.■ Cumulative Review27. State the greater than property.
![Page 48: Lesson 9.3 Arcs pp. 381-387](https://reader035.fdocuments.es/reader035/viewer/2022081506/56814fe5550346895dbdaf58/html5/thumbnails/48.jpg)
■ 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.