1. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
Theorem 6.9 Measure of an Inscribed Angle Theorem
The measure of an inscribed angle is one half the measure of its intercepted arc. D C A B

2. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
Example 1
Use inscribed angles P Q S R T
Find the indicated measure in P.
Solution
Because RQS is a semicircle,

3. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
Checkpoint. Find the indicated measure. F J H G D B C

4. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
Theorem 6.10
If two inscribed angles of a circle intercept the same arc, then the angles are congruent. A B D C

5. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
Theorem 6.11 D A B C
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.
if and only if ____ is a diameter of the circle.

6. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
Theorem 6.12
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. F G E D C
D, E, F, and G lie in C if and only if

7. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
Example 2
Use the circumscribed circle
Security A security camera that rotates 90o is placed in a location where the only thing viewed when rotating is the wall. You want to change the camera’s location. Where else can it be placed so that the wall is viewed in the same way?
Solution
From Theorem 6.11, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a ________ of the circle.
diameter
So, draw the circle that has the width of the wall as a _________. The wall fits perfectly with your camera’s 90 rotation from any point on the _________ in front of the wall.
diameter
semicircle

8. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
Checkpoint. Complete the following exercises. S T R Q
By Theorem 6.10, if two inscribed angles of a circle intercept the same arc, then the angles are congruent.

9. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
Checkpoint. Complete the following exercises.
A right triangle is inscribed in a circle. The radius of the circle is 5.6 centimeters. What is the length of the hypotenuse of the right triangle?
By Theorem 6.11, if a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.

10. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
Example 3
Use Theorem 6.12
Find the value of each variable. R S Q P
Solution
PQRS is inscribed in a circle, so its opposite angles are _____________.
supplementary

11. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
Example 3
Use Theorem 6.12
Find the value of each variable. L M K J
Solution
JKLM is inscribed in a circle, so its opposite angles are _____________.
supplementary

12. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
Checkpoint. Complete the following exercises. D B A C

13. Use Inscribed Angles and Polygons
6.4
Use Inscribed Angles and Polygons
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