1. Do Now

2. 10.4: Use Inscribed Angles and Polygons
Geometry
10.4: Use Inscribed Angles and Polygons

3. Inscribed Angle
An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.
The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle.
BCA is an inscribed angle.
BA is an intercepted arc of
BCA. B W C A

4. Measure of an Inscribed Angle
The measure of an inscribed angle is half the measure of its intercepted arc.
If m BCA = 20o
Then m BA = 40o B W C A

5. Theorem 10.8
If two inscribed angles of a circle intercept the same arc, then the angles are congruent. B
If m BA = 40o
then m BCA = 20o
and m BDA = 20o W C A D

6. Theorem 10.9
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. (The converse is also true.) A B W C

7. Examples

8. Theorem 10.10
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
Angles B and D are supplementary
And
Angles A and C are supplementary B A W D C

9. Examples

10. Do Now: 10.4 Review