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