Inscribed Angles

1. Inscribed Angles Section 10.5

2. Essential Questions • 1. How to use inscribed angles to solve problems in geometry • 2. How to use inscribed angles to solve real-life problems

3. intercepted arc inscribed angle Definitions • Inscribed angle – an angle whose vertex is on a circle and whose sides contain chords of the circle • Intercepted arc – the arc that lies in the interior of an inscribed angle and has endpoints on the angle

4. Measure of an Inscribed Angle Theorem (10.9) • If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

5. Example 1 • Find the measure of the blue arc or angle. a. b.

6. Congruent Inscribed Angles Theorem (10.10) • If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

7. Example 2

8. Definitions • Inscribed polygon – a polygon whose vertices all lie on a circle. • Circumscribed circle – A circle with an inscribed polygon. The polygon is an inscribed polygon and the circle is a circumscribed circle.

9. Inscribed Right Triangle Theorem(10.11) • 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.

10. Inscribed Quadrilateral Theorem(10.12) • A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

11. Example 3 • Find the value of each variable. b. a.

12. Assignment: • P. 508/ 1- 14 • HW: p. 508/ 15- 33 all