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Section 10-4
Inscribed Angles
Thursday, May 17, 2012

Essential Questions
How do you find measures of inscribed angles?
How do you find measures of angles on inscribed polygons?

Vocabulary
1. Inscribed Angle:
2. Intercepted Arc:

Vocabulary
1. Inscribed Angle: An angle made of two chords in a circle, so that
the vertex is on the edge of the circle
2. Intercepted Arc:

Vocabulary
1. Inscribed Angle: An angle made of two chords in a circle, so that
the vertex is on the edge of the circle
2. Intercepted Arc: An arc with endpoints on the sides of an
inscribed angle and in the interior of the inscribed angle

Theorems
10.6 - Inscribed Angle Theorem:
10.7 - Two Inscribed Angles:
10.8 - Inscribed Angles and Diameters:

Theorems
10.6 - Inscribed Angle Theorem: If an angle is inscribed in a circle,
then the measure of the angle is one half the measure of the
intercepted arc
10.7 - Two Inscribed Angles:

Theorems
intercepted arc
10.7 - Two Inscribed Angles: If two inscribed angles of a circle
intercept the same arc or congruent arcs, then the angles are
congruent

Theorems
intercepted arc
10.7 - Two Inscribed Angles: If two inscribed angles of a circle
intercept the same arc or congruent arcs, then the angles are
congruent
10.8 - Inscribed Angles and Diameters: An inscribed angle of a
triangle intercepts a diameter or semicircle IFF the angle is a right
angle

Example 1
Find each measure.
a. m∠YXW
b. mXZ


Example 1
Find each measure.
a. m∠YXW
m∠YXW =
1
2
mYW

b. mXZ


Example 1
Find each measure.
a. m∠YXW
m∠YXW =
1
2
mYW

=
1
2
(86)
b. mXZ


Example 1
Find each measure.
a. m∠YXW
m∠YXW =
1
2
mYW

=
1
2
(86) = 43°
b. mXZ


Example 1
Find each measure.
a. m∠YXW
m∠YXW =
1
2
mYW

=
1
2
(86) = 43°
b. mXZ

mXZ

= 2m∠XYZ

Example 1
Find each measure.
a. m∠YXW
m∠YXW =
1
2
mYW

=
1
2
(86) = 43°
b. mXZ

mXZ

= 2m∠XYZ = 2(52)

Example 1
Find each measure.
a. m∠YXW
m∠YXW =
1
2
mYW

=
1
2
(86) = 43°
b. mXZ

mXZ

= 2m∠XYZ = 2(52) =104°

Example 2
Find m∠QRT when m∠QRT = (12x − 13)° and m∠QST = (9x + 2)°.

Example 2
12x −13= 9x +2

Example 2
12x −13= 9x +2
3x =15

Example 2
12x −13= 9x +2
3x =15
x =5

Example 2
12x −13= 9x +2
3x =15
x =5
m∠QRT =12(5)−13

Example 2
12x −13= 9x +2
3x =15
x =5
m∠QRT =12(5)−13= 60−13

Example 2
12x −13= 9x +2
3x =15
x =5
m∠QRT =12(5)−13= 60−13 = 47°

Example 3
Prove the following.
Given: LO

≅ MN

Prove: MNP ≅LOP

Example 3
Prove the following.
Given: LO

≅ MN

Prove: MNP ≅LOP
There are many ways to prove this one. Work through
a proof on your own. We will discuss a few in class.

Example 4
Find m∠B when m∠A = (x + 4)° and m∠B = (8x - 4)°.

Example 4
m∠A+ m∠B + m∠C =180

Example 4
m∠A+ m∠B + m∠C =180
x +4+8x −4+90 =180

Example 4
m∠A+ m∠B + m∠C =180
x +4+8x −4+90 =180
9x +90 =180

Example 4
m∠A+ m∠B + m∠C =180
x +4+8x −4+90 =180
9x +90 =180
9x = 90

Example 4
m∠A+ m∠B + m∠C =180
x +4+8x −4+90 =180
9x +90 =180
9x = 90
x =10

Example 4
m∠A+ m∠B + m∠C =180
x +4+8x −4+90 =180
9x +90 =180
9x = 90
x =10
m∠B = 8(10)−4

Example 4
m∠A+ m∠B + m∠C =180
x +4+8x −4+90 =180
9x +90 =180
9x = 90
x =10
m∠B = 8(10)−4 = 80−4

Example 4
m∠A+ m∠B + m∠C =180
x +4+8x −4+90 =180
9x +90 =180
9x = 90
x =10
m∠B = 8(10)−4 = 80−4 =76°

Check Your Understanding
p. 713 #1-10

Problem Set

Problem Set
p. 713 #11-35 odd, 49, 55, 61
“You're alive. Do something. The directive in life, the moral imperative
was so uncomplicated. It could be expressed in single words, not
complete sentences. It sounded like this: Look. Listen. Choose. Act.”
- Barbara Hall

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