1. Section 10.3 – Inscribed Angles
Chapter 10 – Circles
Section 10.3 – Inscribed Angles

2. Unit Goal
Use inscribed angles to solve problems.

3. Basic Definitions
INSCRIBED ANGLE – an angle whose vertex is on the circle
INTERCEPTED ARC – the arc whose endpoints are are on the inscribed angle
is an inscribed angle.
is the intercepted arc.

4. What Is the Measure of an Inscribed Circle?

5. Theorem 10.8 Measure of an Inscribed Angle
The measure of an inscribed angle is ½ of its intercepted arc.

6. Example
Find the measure of the angle or arc:
20º

7. Example
Find the measure of the angle or arc:
50º

8. Example
60º

9. Theorem 10.9
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
60º

10. Properties of Inscribed Polygons
If all the vertices of a polygon lie on a circle, the polygon is INSCRIBED in the circles and the circle is CIRCUMSCRIBED about the polygon

11. Theorems About Inscribed Polygons
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 iff segment AC is a diameter of the circle

12. Theorem 10.11
A quadrilateral can be inscribed in a circle iff its opposite angles are supplementary
D, E, F, and G lie on some circle C iff
m<D + m<F = 180° AND
m<E + m<G = 180°

13. Example Substitute this into the second equation 5x + 2y = 180
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 and 5x+ 2y = 180
3x + 3y = 180 (solve for x)
- 3y y
3x = -3y + 180
3 3
x = -y + 60
Substitute
Substitute this into the second equation 5x + 2y = 180
5 (-y + 60) + 2y = 180
-5y y = 180
-3y = -120
y = 40
x = -y + 60
x = = 20

14. Example (cont.)
x = 20, y = 40
m<A = 2y, m<B = 3x, m<C = 5x, m<D = 3y
m<A = 80°
m<B = 60°
m<C = 100°
m<D = 120°