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

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,

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

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

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.

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

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

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.

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.

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

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

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

Use Inscribed Angles and Polygons 6.4 Use Inscribed Angles and Polygons Pg. 218, 6.4 #1-32