1. 10-4 Inscribed Angles The student will be able to:
Find measures of inscribed angles.
Find measures of angles of inscribed polygons.

2. Inscribed Angles
Inscribed Angles – have a vertex on a circle and sides that contain chords of the circle.
In C, is an inscribed angle.
Intercepted arc – has endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle.
In C, minor arc is intercepted by
There are three ways that an angle can be inscribed in a circle.

3. If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc.
and
Example 1:
Find each measure:

4. Related Inscribed Angles
If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent.
and both intercept
So,
Example 2:
If and , find
= 3(8)
= 24°

5. Angles of Inscribed Polygons
An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle.
If is a semicircle, then
If then is a semicircle and
is a diameter.
Example 3:
If and , find x.
ΔFGH is a right triangle because inscribes
a semicircle.
7x x – =
180
24x + 84 = 180
24x = 96
x = 4

6. Quadrilaterals Inscribed in a Circle
Only certain quadrilaterals can be inscribed in a circle.
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
If quadrilateral KLMN is inscribed in A,
then and are supplementary and
and are supplementary.
Example 4:
Quadrilateral WXYZ is inscribed in V. Find
and
x + 95 = 180
x + 60 = 180
x = 85
x = 120

7. You Try It: 1. Find An insignia is and emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find and
x x – = 180
= 8(10) - 4
9x = 90 =
x = 10
= 76
x + 90 = 180
14x + 8x+4 = 180
x = 90
22x = 176
x = 8
= 8(8) + 4
= 68