inscribed angles 10.3 california state standards 4: prove theorems involving congruent and similar...
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Inscribed Angles 10.3
California State Standards4: Prove theorems involving congruent and similar7: Prove/use theorems involving circles.16: Find probability given graph or table
21: Prove/solve relationships with circles.
definitionInscribed Angle
An angle whose vertex is on a circle and whose sides contain chords of the circle.
P
QR
definitionIntercepted Arc
An arc that lies in the interior of an inscribed angle and has endpoints on the angle.
P
QR
definitionsInscribed Polygon
A polygon with all of its vertices on a circle.
CircumscribeTo circle around something.
A
B C
D
EF
P
hexagon is inscribed in
ABCDEFP
circumscribeshexagon
PABCDEF
theoremInscribed Angle Measure
The measure of an inscribed angle is half the measure of its intercepted arc.
C
A
B
12m ACB mAB
D
example
B
C
140o
70o
Find the measure of the arc or angle.
A
mAC
C
theoremCongruent Inscribed Angles
Inscribed anglesare congruent.
B
A
E AAEB CB
that intercept the same arc
m ADE
m ACE
m ABE
example
CB
D
A
E
90o
45o
45o
45o
Find the measure of the arc or angle.
theoremInscribed Right Triangle
An inscribed right triangle has a diameter as its hypotenuse.
C
Q
P
R
is a right trianglePQR is a diameterPR
example
B
DA
C
Find the measure of the arc or angle.
180omADC
theoremInscribed Quadrilateral
A quadrilateral can be inscribed in a circle if and only if its opposite angles are
supplementary.
S
Q
P
R
180
180
m P m R
m Q m S
85 180y 95y
100x 80 180x
example
TS
R85o
U80o
xo yo
Find value of x.
example
B
A
C98o
196o
Find the measure of the arc or angle.
m ABC
D
m C
example
CB
D
44o44o
Find the measure of the arc or angle.
A
example
C
B
A
3xo
123 (180)x
30x
Find value of x.
_ _ _220 190 1800760 190 3420
x yx y
22 19 18040 10 180
x yx y
example
B
A
D
22xo
C40xo10yo
19yo
Find value of the measure of each angle.
3x 540 1620x
22(3) 66m D
40(3) 120m C
example
B
A
D
22xo
C40xo10yo
19yo
Find value of the measure of each angle.
180 114m B m D
180 60m A m C
120o
66o
60o
114o