Chapter 10, Section 3:
Inscribed Angles
How can we apply properties of inscribed angles to help us choose a seat for the Hunger Games movie or Snow White with Julia Roberts?
Section 10.3 Goals
G-C.2
1. I can define inscribed angles and apply their properties to solve problems.
2. I can apply properties of inscribed polygons to solve problems.
1

Using Inscribed Angles inscribed angle intercepted arc inscribed angle an angle whose vertex is on a circle and whose sides contain chords of the circle.
intercepted arc - the arc that lies in the interior of an inscribed angle and has endpoints on the angle.
If an angle is inscribed in a circle, then its  measure is one half the measure of its  intercepted arc.
m ∠ ADB = ½m
2

Find the measure of the blue arc or angle.
m         = 2m
∠
QRS =
2(90°) = 180°
Ex. 2:  Finding Measures of Arcs and
Inscribed Angles
Find the measure of the blue arc or angle.
m
∠
ZYX = 115°
3

Find m
∠
ACB, m
∠
ADB, and m
∠
AEB.
Using Theorem 10.8 we can say that the measure of each angle is half  the measure of      .  If m       = 60°,  the measure of each angle is 30°
Theorem 10.9
If two inscribed angles of a circle intercept the  same arc, then the angles are congruent.
∠ C  ≅   ∠ D m
∠
C = 75 °
Both
∠ C and  ∠ D  intercept
So...
4

When you go to the movies,  you want to be close to the  movie screen, but you don’t  want to have to move your  eyes too much to see the  edges of the picture.
If E and G are the ends of the screen and you  are at F, m
∠
EFG is called your viewing angle.
You decide that the middle of the sixth row has  the best viewing angle.  If someone else is  sitting there, where else can you sit to have the  same viewing angle?
SHOW ME
Solution:  Draw the circle that is  determined by the endpoints of the screen  and the sixth row center seat.  Any other  location on the circle will have the same  viewing angle.   Theorem 10.9
5

Homework
Textbook Page 617, #10 - 20 even
If all the vertices of a polygon lie on a circle, the polygon is said to be inscribed and the circle is said to be circumscribed.
6

Theorem 10.10
If a right triangle is inscribed in a circle, then the  hypotenuse is a diameter of the circle.
Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.
∠
B is a right angle if and only if AC is a diameter  of the circle.
Find the value of  each variable.
AB is a diameter.   So,
∠
C is a right angle and m
∠
C = 90 °
2x° = 90° x = 45
2x°
Theorem 10.11
A quadrilateral can be  inscribed in a circle if and only if its opposite angles are supplementary.
D, E, F, and G lie on  some circle, C, if and only if m
∠
D + m
∠
F = 180 ° and m
∠
E + m
∠
G = 180 °
Find the value of  each variable.
DEFG is inscribed in  a circle, so opposite angles are supplementary.
m
∠
E + m
∠
G = 180° y + 120 = 180 y = 60 y° z°
120°
80°
7

3y°
2y°
3x°
In the diagram, ABCD is  inscribed in circle P.
Find the measure of each angle.
ABCD is inscribed in a  circle, so opposite angles are supplementary.
3x + 3y = 180
5x + 2y = 180
2x°
To solve this system of linear equations, you can solve the first  equation for y to get y = 60 ­ x.  Substitute this expression into  the second equation.
Solution
Solutions: x = 20 and y = 40, so m
∠
A = 80°, m
∠
B = 60°, m
∠
C
= 100°, and m
∠
D = 120°
8

Arcs and Angles
Graphic Organizer
Graphic Organizer: Review
Using the figure to the right, name one example for each of the  following:
1.      major arc
2.     inscribed angle
3.     central angle
4.     intercepted arc
5.     semicircle
6.     minor arc
9

Section 10.1 - 10.3 Review
10