Download 10.4 Inscribed Angles and Polygons

10.4 Inscribed
Angles and Polygons
Inscribed Angle:
• An angle whose vertex is on a circle and
whose sides contain chords of the circle.
O
<OMP is an inscribed angle.
P
N
Intercepted Arc:
The arc inside the inscribed angle;
its endpoints are on the angle.
M
^OP is the intercepted arc.
It’s the part of the circle cut off by the angle.
The measure of an inscribed angle:
• One half the measure
of its intercepted arc.
O
m OP on
NM = 66.01
(Let’s just say 66.)
P
What is the measure of angle OMP?
N
m OMP = 33.00
What is the measure of angle ONP?
M
66
Inscribed Polygon:
• Polygon having all vertices as points of the circle.
circumscribed
• The circle around it is ________________.
B
A
ABCDE
____________is
inscribed.
P
Circle P
____________is
circumscribed.
E
C
D
EXAMPLE 1
Use inscribed angles
Find the indicated measure in
a.
m
T
b.
P.
mQR
SOLUTION
a. M
T = 1 mRS = 1 (48o) = 24o
2
2
b. mTQ = 2m
R = 2 50o = 100o. Because TQR is a semicircle,
mQR = 180o– mTQ = 180o – 100o = 80o. So, mQR = 80o.
EXAMPLE 2
Find the measure of an intercepted arc
Find mRS and m
and RUS?
STR. What do you notice about
STR
SOLUTION
From Theorem 10.7, you know that mRS = 2m
= 2 (31o) = 62o.
Also, m
STR =
1
1
mRS = (62o) = 31o. So,
2
2
RUS
STR RUS.
What is the measure of angle B?
What is the measure of angle D?
m AC on
A
Can you make any conclusions about
Inscribed angles that intersect the
same arc?
ED = 54.14
C
They are both 27.07 .
If two inscribed angles
of a circle intercept the
same arc, then the
angles
are congruent.
B
D
EXAMPLE 3
SOLUTION
Notice that JKM and JLM intercept the same arc, and
so JKM JLM by Theorem 10.8. Also, KJL and KML
intercept the same arc, so they must also be congruent.
Only choice C contains both pairs of angles.
So, by Theorem 10.8, the correct answer is C.
If IF is a diameter, what
is the measure of angle G?
m IGF = 90.00
I
If a triangle is
inscribed in a circle
so that its side is a
diameter, then the
triangle is a right
triangle.
F
G
A quadrilateral can be
inscribed in a circle
if and only if its opposite
angles are supplementary.
Q
P
Which angles are supplementary?
N
M
What is the sum of all of the angles?
O
Q, O
and M, P
360
EXAMPLE 5
Use Theorem 10.10
Find the value of each variable.
a.
SOLUTION
a.
PQRS is inscribed in a circle, so opposite angles
are supplementary.
m
P+m
R = 180o
75o + yo = 180o
y = 105
m
Q+m
S = 180o
80o + xo = 180o
x = 100
EXAMPLE 5
Use Theorem 10.10
Find the value of each variable.
b.
SOLUTION
b.
JKLM is inscribed in a circle, so opposite angles
are supplementary.
m K + m M = 180o
m J + m L = 180o
2ao + 2ao = 180o
4bo + 2bo = 180o
4a = 180
6b = 180
a = 45
b = 30
Geometry
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Page 676 (1,2,4-7,9-12,16-18,43-47)
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day 2
page 676(13-15,19-25,29,40-42)
Sophomore Math
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Page 676 (1-15)