Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the...

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Geometry Geometry 9.3 Arcs and Central 9.3 Arcs and Central Angles Angles

Transcript of Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the...

Page 1: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

GeometryGeometry

9.3 Arcs and Central Angles9.3 Arcs and Central Angles

Page 2: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

Central AnglesCentral Angles

An angle with the vertex at the An angle with the vertex at the center of the circle.center of the circle.

B

X

Y

A

Q

AQX, AQB, and YQX are examples of central angles.

7 7 7

Page 3: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

ArcArc

An unbroken part of the circle.An unbroken part of the circle.

B

X

Y

A

Q

AB

XBA

Page 4: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

Measures of an arcMeasures of an arc

B

X

Y

A

Q

AX

Minor ArcsHas a measure between 0 and 180 degrees.Needs only two letters in its symbol.

B

X

Y

A

Q

AXY

Major ArcsHas a measure between 180 and 360 degrees.Needs three letters in its symbol.

B

X

Y

A

Q

XBY

SemicircleHas a measure of 180 degrees.Needs three letters in its symbol.

The measure of a minor arc is equal

to the measure of its central angle.

Please put minor arc, major arc, and semicircle in the same box on your Vocab List!!!

Page 5: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

Using the letters shown in the diagram, name:

1. four central angles

2. two semicircles

3. four minor arcs

4. four major arcs

W

X

YQ

Z

7 7 77WQX XQY YQZ XQZ

WXY XYZ

WX YX ZY WZ

WXZ WZX YZX ZXYAre these the same?

Page 6: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

Adjacent ArcsAdjacent Arcs

Arcs with exactly one point in Arcs with exactly one point in common.common.

J

K

I

Are arcs that overlap adjacent?

No, because they would have more than one common point.

IJ and JK are adjacent arcs.

Page 7: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

Arc Addition PostulateArc Addition Postulate

The measure of the arc formed by two adjacent arcs is The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs.the sum of the measures of these two arcs.

mBC + mCD = mBCD

D

C

B

A

Find the mistake on your handout.

Minor arc only needs two letters.

Page 8: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

Find each measure.

QP

ST

C

45

60

5.

8. SQ

11. SPQ

14. SPT

6. ST

9.

12. PT

15.

7. SQP

10.

13.

PCQ

SCQ SCP

TCP

TSQ

60o 45o

120o

180o

120o 120o 180o

240o

135o

135o 135o

360 – 45 = 315o 97.5o

Page 9: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

Find the measure of each numbered angle. O is the center of the circle.

240 40

1O 21

OO

1O

2

1

16. 17. 18. 19.

120o

60o

m 1 = 180o – m 2

m 2 = 180o – m 1

7 7

77

40o

140o

Page 10: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

Congruent ArcsCongruent Arcs

Arcs in the same circle or congruent circles Arcs in the same circle or congruent circles that have equal measures are congruent.that have equal measures are congruent.

XY = AB

RY = QA ≠ SPR

XY

C

B

A

Q S P

T

~

but neither arc is congruent to ST because circle P is not congruentto the other two circles.

Page 11: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

TheoremTheorem

In the same circle or in congruent circles, two In the same circle or in congruent circles, two minor arcs are congruent if and only if their minor arcs are congruent if and only if their central angles are congruent.central angles are congruent.

If m 1 = m 2, then JK = LM. L

M

J

21

K

7 7 ~

If JK = LM, then m 1 = m 2.

7 7~

Page 12: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

• The figure shows two concentric circles with center N. Classify

each statement as true of false

45mBC AB VW

90m DNC 45mXY

VW WX AED VZY

20. 21.

22. 23.

24. 25.

45

True False

True False

True

N X

W

V

A

C

B

D

Y

E

Z

False

True/False: mAB = mVW

True

Page 13: Geometry 9.3 Arcs and Central Angles. Central Angles An angle with the vertex at the center of the circle. An angle with the vertex at the center of the.

HWHW

P. 341-342 WE 1-11, 16-18P. 341-342 WE 1-11, 16-18

for 17-18 see example P. 340for 17-18 see example P. 340

Note-do constructions 8-10 during Note-do constructions 8-10 during this chapterthis chapter


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