Geometry Section 10-4 1112

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. m XZ


Example 1
Find each measure.
a. m∠YXW

1 
m∠YXW = mYW
2


b. m XZ


Example 1
Find each measure.
a. m∠YXW

1  1
m∠YXW = mYW = (86)
2 2


b. m XZ


Example 1
Find each measure.
a. m∠YXW

1  1
m∠YXW = mYW = (86) = 43°
2 2


b. m XZ


Example 1
Find each measure.
a. m∠YXW

1  1
m∠YXW = mYW = (86) = 43°
2 2


b. m XZ

m XZ = 2m∠XYZ


Example 1
Find each measure.
a. m∠YXW

1  1
m∠YXW = mYW = (86) = 43°
2 2


b. m XZ

m XZ = 2m∠XYZ = 2(52)


Example 1
Find each measure.
a. m∠YXW

1  1
m∠YXW = mYW = (86) = 43°
2 2


b. m XZ

m XZ = 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

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was so uncomplicated. It could be expressed in single words, not
complete sentences. It sounded like this: Look. Listen. Choose. Act.”
- Barbara Hall

Geometry Section 10-4 1112

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