Find the measure of each lettered angle.

Warm up

UNIT OF STUDYLesson 6.3

ARCS AND ANGLES

TOPIC VII - CIRCLES

Warm up = (FRI Starting Point)

You will learn ….(FRI Ending Point)

To discover relationships

between an inscribed angle of a

circle and its intercepted arc

Content….(FRE - Research)

Complete your investigations, and as

you see the power point presentation,

on your notebook and/or your packet,

write down the key concepts and

complete the conjectures.

Central AnglesAngle whose vertex is at the center of a circle.

AOB is a central angle of circle O

A

D

B

Inscribed Angle

Angle that has its vertex on the circle and its sides are chords.

ABC is an inscribed angles of circle O

A

B

O

D

O

Two types of angles in a circleARCS AND ANGLES

ARCS AND ANGLES

It is two points on the circle and the continuous (unbroken) part of the circle between the two points.

A

o

Example: AB

Minor arc is an arc that is smaller than a semicircle and are named by their end points

Major arc is an arc that is larger than a semicircle and are named by their end points and a point on the arc C

Example: ABC

B

Arc Definition

ARCS AND ANGLES

C

o

m CAR = ½ m COR

The measure of an angle inscribed in a circle is one-half (1/2) the measure of the intercepted arc.

A

R

Inscribed Angle Properties

100o

50o

Incribed Angle Conjecture

ARCS AND ANGLES

A

AQB APB

Inscribed angles that intercept the same arc are congruent

P

B

Inscribed Angle Intercepting the same arc

80o

Incribed Angle intercepting Conjecture

Q

80o

ARCS AND ANGLES

A

Angles inscribed in a semicircle are right angles

B

Angles Inscribed in a Semicircle

Angle Inscribed in a semicircle Conjecture90o

90o

90o

ARCS AND ANGLES

A

The opposite angles of a cyclic quadrilateral are supplementary

B

Cyclic Quadrilaterals

Cycle quadrilaterals Conjecture

101o

132o

79o

48o

ARCS AND ANGLES

By the Cyclic Quadrilateral Conjecture,

w +100° = 180°, so w = 80°.

PSR is an inscribed angle for PR.

m PR = 47°+73° = 120°, so by the Inscribed Angle Conjecture,

x = ½(120°) =60°.

By the Cyclic Quadrilateral Conjecture,

x + y = 180°.

Substituting 60° for x and solving the equation gives y =120°.

By the Inscribed Angle Conjecture, w = ½ (47° + z).

Substituting 80° for w and solving the equation gives z = 113°.

Cyclic Quadrilaterals

Find each lettered measure

Find each lettered measure.

Cyclic Quadrilaterals

ARCS AND ANGLES

A

Parallel lines intercept congruent arcs on a circle.

B

Arcs by Parallel lines

Parallel lines intercepted Arcs Conjecture

secant

D

C

AD BC

Practice (FRI Skill development)

Arcs Length

TOPIC VII - CIRCLES

ARCS LENGTH

It is two points on the circle and the continuous (unbroken) part of the circle between the two points.

Minor arc is an arc that is smaller than a semicircle and are named by their end points

Arc Definition

C

o

m COR = 100o

The measure of the minor arc is the measure of the central angle.

A

R

100o

CR = 100o

The measure of the arc from 12:00 to 4:00 is equal to the measure of the angle formed by the hour and minute hands

A circular clock is divided into 12 equal arcs, so the measure of each hour is

360 or 30°. 12

ARCS LENGTH

Because the minute hand is longer, the tip of theminute hand must travel farther than the tip of the hour hand even though they both move 120° from 12:00 to 4:00.

So the arc length is different even though the arc measure is the same!

ARCS LENGTH

The arc measure is 90°, a full circle measures 360°, and 90° = 1. 360° 4

The arc measure is half of the circlebecause 180° = 1 360° 2

The arc measure is one-third of the circle because 120° = 1 360° 3

The arc length is some fraction of the circumference of its circle.

ARCS LENGTH

ARCS LENGTH

The arc measure is 90°, a full circle measures 360°, and 90° = 1. 360° 4

The arc measure is half of the circle because 180° = 1 360° 2

To find the arcs length we have to follow this stepsStep 1: find what fraction of the circle each arc

For AB and CED find what fraction of the circle each arc is

ARCS LENGTH

Circle TC = 2(12 m)C= 24 m

Circle O C= 2 (4 in.)C= 8 in

Step 2: Find the circumference of each circle

Step 3: Combine the circumferences to find the length of the arcs

Circle T Length of AB= 90° 2 (12m) 360°Or AB= 90° 24 m 360°

AB = 18.84 m

Circle O Length of CD= 180° 2 (4 in) 360°Or CD= 180° 8 in 360°

CD = 12.56 in

Step 1: find what fraction of the circle the arc is

The length of an arc equals the measure of the arc divided by 360° times the circumference

ARCS LENGTHArc Length conjecture

l = x . 2 r3600

ARCS LENGTH

Remember:

The arc is part of a circle and its length is a part of the circumference of a circle.

The measure of an arc is calculated in units of degrees, but arc length is calculated in units of distance (foot, meters, inches, centimeter.

ARCS LENGTHExample: