Lesson 6.3 Inscribed Angles and
their Intercepted Arcs
Goal 1
Using Inscribed Angles
Goal 2
Using Properties of Inscribed
Angles.
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Using Inscribed Angles
Inscribed Angles & Intercepted Arcs
An INSCRIBED ANGLE is an angle
whose vertex is on the circle and whose
sides each contain chords of a circle.
B
A
D
C
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Using Inscribed Angles
Measure of an Inscribed Angle
If an angle is inscribed in a circle, then the measure of the
angle equals one-half the measure of its intercepted arc.
1
m  = 2 m arc
OR
2 m  = m arc
B
B
B
50°
x°
50°
B
50°
A
C
A
C
100°
100°
A
C
100°
A
C
2x°
3

Using Inscribed Angles
Example 1:
Find the m PQ and mPAQ .

63
PQ =2 * m PBQ
= 2 * 63
= 126˚
mPAQ = m PBQ
mPAQ = 63˚
4
Using Inscribed Angles
Example 2:
Find the measure of each arc or angle.

Q = ½ 120 = 60˚
QSR = 180˚
R = ½(180 – 120)
= ½ 60
= 30˚
5
Using Inscribed Angles
Inscribed Angles Intercepting Arcs Conjecture
If two inscribed angles intercept the same arc or
arcs of equal measure then the inscribed angles
have equal measure.
D
A
mCAB = mCDB
P
C
B
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
Using Inscribed Angles
Example 3:


F
Find m EDF
E
70
A
D
m EF  2 * 70  140
m EDF =360 – 140 = 220˚
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
Using Properties of Inscribed Angles
Example 4:
mCAB = ½

Find mCAB and m AD
CB
C
60°
mCAB = 30˚


B
m AD = 2* 41˚
m AD = 82˚
P
A
41°
D
8
Using Properties of Inscribed Angles
Cyclic Quadrilateral
A polygon whose vertices lie on the circle,
i.e. a quadrilateral inscribed in a circle.
B
Quadrilateral ABFE is
inscribed in Circle O.
A
O
F
E
9
Using Properties of Inscribed Angles
Cyclic Quadrilateral Conjecture
If a quadrilateral is inscribed in a
circle, then its opposite angles are
supplementary.
10
Using Properties of Inscribed Angles
Circumscribed Polygon
A polygon is circumscribed about a
circle if and only if each side of the
polygon is tangent to the circle.
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Using Inscribed Angles
Example 5:
F
Find mEFD
E
D
A
B
mEFD = ½ 180 = 90˚
12
Using Properties of Inscribed Angles
Angles inscribed in a Semi-circle Conjecture
A triangle inscribed in a circle is a
right triangle if and only if one of its
sides is a diameter.
A has its vertex on the
circle, and it intercepts
half of the circle so that
mA = 90.
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
Using Properties of Inscribed Angles
Example 6:
Find the measure of GDE
Find x.
3x° F
E
A
D
C
B
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Using Properties of Inscribed Angles
Find x and y
y°
x°
(2y - 3)°
80°
85°
3x°
(y + 5)°
15
Using Properties of Inscribed Angles
Parallel Lines Intercepted Arcs Conjecture
Parallel lines intercept congruent arcs.
X
A
Y
B
16
Using Properties of Inscribed Angles
Find x.
360 – 189 – 122 = 49˚
x
x = 49/2 = 24.5˚
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Homework:
Lesson 6.3/ 1-14
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