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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
An INSCRIBED ANGLE is an angle whose vertex is on the circle and whose sides each contain chords of a circle.
Inscribed Angles & Intercepted Arcs
D
B A
C
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Using Inscribed Angles
If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc.
m = m arc OR 2 m = m arc
Measure of an Inscribed Angle
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1
50°
100°
B
AC
50°
100°
B
AC
50°
100°
B
AC
x°
2x°
B
AC
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Using Inscribed Angles
Example 1:
63
Find the m and mPAQ .PQ
mPAQ = m PBQmPAQ = 63˚
PQ =2 * m PBQ
= 2 * 63 = 126˚
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Using Inscribed Angles
Find the measure of each arc or angle.
QSR
Example 2:
Q
R
= ½ 120 = 60˚
= 180˚
= ½(180 – 120)= ½ 60= 30˚
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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.
mCAB = mCDB
P
A
BC
D
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Using Inscribed Angles
Example 3:
70E D
A
FEDFmFind
14070*2 EFm
EDFm =360 – 140 = 220˚
m = 82˚
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Using Properties of Inscribed Angles
Example 4:
41°
60°
P
C
DA
B
Find mCAB and m AD
mCAB = ½
mCAB = 30˚ADm = 2* 41˚ AD
CB
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Using Properties of Inscribed Angles
Cyclic QuadrilateralA polygon whose vertices lie on the circle, i.e. a quadrilateral inscribed in a circle.
Quadrilateral ABFE is inscribed in Circle O.
O
A
B
F
E
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Using Properties of Inscribed Angles
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
Cyclic Quadrilateral Conjecture
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Using Properties of Inscribed Angles
A polygon is circumscribed about a circle if and only if each side of the polygon is tangent to the circle.
Circumscribed Polygon
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Using Inscribed Angles
E DA
B
FExample 5:
Find mEFD
mEFD = ½ 180 = 90˚
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Using Properties of Inscribed Angles
A triangle inscribed in a circle is a right triangle if and only if one of its
sides is a diameter.
Angles inscribed in a Semi-circle Conjecture
A has its vertex on the circle, and it intercepts half of the circle so thatmA = 90.
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Using Properties of Inscribed Angles
Find the measure ofGDE
Example 6:
Find x. 3x°E
D
A
B
C
F
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Using Properties of Inscribed Angles
Find x and y
3x°
(y + 5)°
(2y - 3)°
85°80°
y°x°
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Using Properties of Inscribed Angles
Parallel Lines Intercepted Arcs Conjecture
Parallel lines intercept congruent arcs.
A
B
X
Y
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Using Properties of Inscribed Angles
Find x.
x122˚
189˚
360 – 189 – 122 = 49˚
x = 49/2 = 24.5˚
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Homework:
Lesson 6.3/ 1-14
