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Math-2Lesson 6-2

Exterior Angle Theorem,

Arcs, Central Angles, and

Inscribed Angles in Circles

Exterior angle: An angle formed by one side of a triangle

and the extension of the adjacent side of the triangle.

A

B

C E

Angle “E” is an exterior angle to triangle ABC.

Remote interior angle: The two angles of a triangle that are on opposite sides of the triangle from the exterior

angle.

Angles “A” and “B” are “remote interior” angles to

exterior angle “E”.

What two theorems would be needed to prove that the

measure of an “exterior angle” equals the sum of the

remote interior angles?

A

B

C E

EmBmAm

The “exterior angle” theorem

A

B

C E

180 CmBmAm theorem"sum Triangle"

180 EmCm Pairs"Linear "

EmCmCmBmAm on"substituti"

EmBmAm

)left/right from Cm(subtract

equality ofProperty

angles.interior remote theof sum theequals

angleexterior an of measure The

QED

Angles “inscribed” on a circle.

Inscribed angle: has its vertex on the circle.

“Central” Angle of a circle.

Central angle: has its vertex at the center of the circle.

“intercepted” Arc

Intercepted arc: the arc of the circle that is in the interior of the angle. It has the same measure as the central angle.

B

D

C

Naming Arcs

The arc subtended by Center Angle C is BDSpoken: “arc BD”

mBDSpoken: “the measure of arc BD”

B

D

C

Minor Arcs: arcs that are less than half the circle.

Major Arcs: arcs that are more than half the circLe.

BED

(minor arc)BCD

To distinguish between minor arc BD and major arc BD,

we could add a letter between ‘B’ and ‘D’ to indicate a

point in between that the arc passes through.

E(major arc)

There is a relationship between the measure of the “Central” Angle and the measure of the “inscribed”

angle that intercepts the same arc.

A

B

C

D

1. Draw a diameter through point A and Center C.

A

B

C

D

2. Two triangles are formed

3. On the circle, segment CA is called a _______.

A

B

C

D

AB

C

D

C

A

4. On the circle, segment CB is called a _______.

5. On the circle, segment CD is called a _______.

6. All radii of a circle are congruent.

A

B

C

D

AB

C

D

C

A

CDACCACB

Since two legs sides of each

triangle are congruent, both

triangles are Isosceles.

7. Base angles of Isosceles Triangles are congruent.

A

B

C

D

xy

y

x

We cannot say the triangles are congruent, however.

8. The measure of angle BCM = _______

A

B

D

xy

y

x

MC

Hint: Use the Exterior Angle Theorem.

2x

x2

9. The measure of angle DCM =______

A

B

D

xy

y

x

x2M

C

2y

y2

10. The measure of inscribed angle BAD = ______

A

B

D

xy

y

x

C x2

y2

x + y

Hint: Use the Angle Addition Postulate

11. The measure of Central angle BCD = ______

A

B

D

yx

y

x

x2Cy2

2x + 2y

)(2 yxBCDm

A

B

D

yx

y

x

)(2 yx C

Can you state theInscribed/Center Angle/Inscribed Arc Theorem?

Inscribed/Center Angle/Inscribed Arc Theorem

A

B

D

x x2C x2

If an inscribed angle and a central angle subtend the same arc, then the measure of the central angle equals twice the measure of the inscribed angle.

If a central angle

subtends an arc, then

the measure of the arc

equals twice the

measure of the

inscribed angle.

Find the measure of the angle.

52

5155180 Nm

?NLMm

?Nm

1. Triangle

2. Inscribed Angle.

To solve for an unknown value, you need an ___________.

Triangle Sum Theorem

Inscribed/Central Angle/Inscribed Arc

Theorem

51

74Nm

Find the measure of the angle.

C

360)( Circlem

110360

110

1. Inscribed Angle.

To solve for an unknown value, you need an ___________.

Inscribed/Central Angle/Inscribed Arc

Theorem

?mFQH 110)55(2 ?Qm

?mFGH

250

250*2

1Qm

125Qm

55

An interesting (and useful) result

F

H

GQ 125

Inscribed angles that “cut opposite arcs are supplementary

(add up to 180).

Find the measure of the angle.

C

125?mFG 250)125(2

?mFGH?Qm

125

mFG mGHmFGH

160250

90

90

An interesting (and useful) result:

Segment QG is a diameter of circle C.

F

GQ

180

C

?mFQH?Fm

An inscribed angle that “cuts a diameter”

always has a measure of 90.

C

1605060360) ( FGarcm

60

90)G ( Farcm

9050)G ( QFarcm

140)G ( QFarcm

70G QHm