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10.4 Inscribed Angles
Objectives


Find measures of inscribed angles
Find measures of angles of inscribed
polygons
Inscribed Angles

An inscribed angle is an angle that
has its vertex on the circle and its
sides are chords of the circle.
A
C
B
Inscribed Angles

Theorem 10.5
A
(Inscribed Angle
Theorem):
The measure of an
inscribed angle equals
½ the measure of its
intercepted arc (or the
measure of the
intercepted arc is twice
the measure of the
inscribed angle).
C
B
mACB = ½m
2 mACB =
or
Example 1:
In
and
Find the measures of the numbered angles.
Example 1:
First determine
Arc Addition
Theorem
Simplify.
Subtract 168 from
each side.
Divide each side
by 2.
Example 1:
So,
m
Example 1:
Answer:
Your Turn:
In
and
measures of the numbered angles.
Answer:
Find the
Inscribed Angles

Theorem 10.6:
If two inscribed s intercept  arcs or the
same arc, then the s are .
mDAC  mCBD
Example 2:
Given:
Prove:
Example 2:
Proof:
Statements
Reasons
1.
1. Given
2.
2. If 2 chords are , corr.
minor arcs are .
3.
3. Definition of
intercepted arc
4.
4. Inscribed angles of
arcs are .
5.
5. Right angles are
congruent
6.
6. AAS
Your Turn:
Given:
Prove:
Your Turn:
Proof:
Statements
Reasons
1.
1. Given
2.
2. Inscribed angles of
arcs are .
3.
3. Vertical angles are
congruent.
4.
4. Radii of a circle are
congruent.
5.
5. ASA
Example 3:
PROBABILITY Points M and N are on a circle so
that
. Suppose point L is randomly located
on the same circle so that it does not coincide with
M or N. What is the probability that
Since the angle measure is twice the arc measure,
inscribed
must intercept
, so L must lie
on minor arc MN. Draw a figure and label any
information you know.
Example 3:
The probability that
is the same as the
probability of L being contained in
.
Answer: The probability that L is located on
is
Your Turn:
PROBABILITY Points A and X are on a circle so
that
Suppose point B is randomly
located on the same circle so that it does not
coincide with A or X. What is the probability that
Answer:
Angles of Inscribed Polygons

Theorem 10.7:
If an inscribed 
intercepts a
semicircle, then the 
is a right .
i.e. If AC is a
diameter of , then
the mABC = 90°.
o
Angles of Inscribed Polygons

Theorem 10.8:
If a quadrilateral is
inscribed in a , then its
opposite s are
D
supplementary.
A
B
O
i.e. Quadrilateral ABCD
is inscribed in O, thus
A and C are
supplementary and B
and D are
supplementary.
C
Example 4:
ALGEBRA Triangles TVU and TSU are inscribed in
with
Find the measure of each
numbered angle if
and
Example 4:
are right triangles.
since
they intercept congruent arcs. Then the third angles of
the triangles are also congruent, so
.
Angle Sum Theorem
Simplify.
Subtract 105 from each side.
Divide each side by 3.
Example 4:
Use the value of x to find the measures of
Given
Answer:
Given
Your Turn:
ALGEBRA Triangles MNO and MPO are inscribed
in
with
Find the measure of each
numbered angle if
and
Answer:
Example 5:
Quadrilateral QRST is inscribed in
find
and
Draw a sketch of this situation.
If
and
Example 5:
To find
To find
we need to know
first find
Inscribed Angle Theorem
Sum of angles in circle = 360
Subtract 174 from each side.
Example 5:
Inscribed Angle Theorem
Substitution
Divide each side by 2.
To find
find
we need to know
but first we must
Inscribed Angle Theorem
Example 5:
Sum of angles in circle = 360
Subtract 204 from each side.
Inscribed Angle Theorem
Divide each side by 2.
Answer:
Your Turn:
Quadrilateral BCDE is inscribed in
find
and
Answer:
If
and
Assignment


Geometry
Pg. 549 #8 – 10, 13 – 16, 18 – 20
Pre-AP Geometry
Pg. 549 #8 – 10, 13 – 20