2. Topic 12-3

3. Definition Secant – a line that intersects a circle in two points.

4. 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 inscribed angle intercepted arc

5. Example 1: Name ALL of the inscribed angles and their corresponding intercepted arcs below. Inscribed angles/Intercepted Arc: ____________________________ ____________________________ ____________________________

6. Central Angle

7. Measure of an Inscribed Angle Theorem If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

8. Example 2: Given that m BC = 100 , find the value of ‘x’ in circle O x = _______________

9. Try This!

10. Try this!

11. Congruent Inscribed Angles Theorem If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

12. Example 3: In circle Q, m ST = 68 . Find the m  1 and m  2. m  1 = _______________ m  2 = _______________

13. 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.

14. Inscribed Right Triangle Theorem If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. The triangle is a right triangle and the angle opposite the diameter is the right angle.

15. Example 5: In circle A, m  1 = (6x + 11) , m  2 = (9x + 19) , m  3 = (4y – 25) , m  4 = (3y – 9) , and PQ  RS. Find m  1, m  2, m  3, m  4 m  1 = _______________ m  2 = _______________ m  3 = _______________ m  4 = _______________

16. Inscribed Quadrilateral Theorem Opposite angles of an inscribed quadrilateral are supplementary.

17. Example 6: Quadrilateral QRST is inscribed in circle C. If m  T = 95 , m  S = 100 , find m  Q and m  R m  Q = __________ m  R = __________

18. Example 3 Find the value of each variable. a. b.