10'3 Inscribed angles

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10.3 Inscribed angles

• Pg 613

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Definitions

• Inscribed angle- an ? whose vertex is on a circle
  and whose sides contain chords of the circle.
• Intercepted arc- the arc that lies in the
  interior of the inscribed angle and has its
  endpts on the angle

B
?BCD
?
BD
(
A
?
C
D
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Thm 10.8measure of an inscribed ?

• If an ? is inscribed in a circle, then its
  measure is ½ the measure of its intercepted arc.
• m?BAC ½ m BC

B
A
C
(
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Example
B
(
m AED ?
180o
D
C
A
E
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Example
A
m?ABD ?
98o
196o
C
B
D
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Example
x ?
70 ½ 7x Or 7x 140
B
70o
A
x20
C
7xo
(
m AD ?
140o
D
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Thm 10.9

• If two inscribed angles of a circle intercept the
  same arc, then the ?s are _at_.

W
Y
X
C
?W ? ?Y
Z
8
Inscribed Polygon
ABCD is inscribed in the circle.

• Polygon with ALL vertices on a circle.

Circumscribed Circle the circle around the
inscribed polygon.
B
A
C
D
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Thm 10.10

• If a rt. ? is inscribed in a circle, then the
  hypotenuse is a diameter of the circle.
• If one side of an inscribed ? is a diameter of
  the circle, then the ? is a rt. ? the ?
  opposite the diameter is the rt. ?.

C
If AC is a diameter, then ?ABC is a rt. ? AND ?B
is the rt. ?.
B
AC is a diameter.
A
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Ex find x.
P

• PQ is a diameter,
• ? ?R is a rt. ?.
• 3x 900
• x 30

R
3xo
C
Q
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Thm 10.11

• A quadrilateral can be inscribed in a circle iff
  its opposite ?s are supplementary.

A
m?A m?C 180o m?B m?D 180o
D
B
So, can a rectangle be inscribed in a circle?
Yes, because its opposite ?s are supplementary.
C
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Ex x? and y?

• 85 x 180
• x 95
• 80 y 180
• y 100

yo
xo
85o
80o
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Ex x? y?
4xy18 y18-4x
Think back to Algebra!
y18-4(3) y18-12 y6
19y
10y

• 40x10y180
• 22x19y180
• 22x19(18-4x)180
• 22x342-76x180
• 342-54x180
• -54x-162
• x3

22x
40x
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Assignment