8-5 Angles in Circles

1. 8-5 Angles in Circles

2. Central Angles • A central angle is an angle whose vertex is the CENTER of the circle NOT A Central Angle (of a circle) Central Angle (of a circle) Central Angle (of a circle)

3. CENTRAL ANGLES AND ARCS The measure of a central angle is equal to the measure of the intercepted arc.

4. Y 110 110 O Z CENTRAL ANGLES AND ARCS The measure of a central angle is equal to the measure of the intercepted arc. Central Angle Intercepted Arc

5. B C 25 A z x y 55 O D EXAMPLE • Segment AD is a diameter. Find the values of x and y and z in the figure. x = 25° y = 100° z = 55°

6. SUM OF CENTRAL ANGLES The sum of the measures fo the central angles of a circle with no interior points in common is 360º. 360º

7. Find the measure of each arc. D C 2x-14 4x 2x 3x E B 3x+10 4x + 3x + 3x + 10+ 2x + 2x – 14 = 360 … x = 26 104, 78, 88, 52, 66 degrees A

8. Inscribed Angles An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords. 3 2 1 4 Is NOT! Is SO! Is NOT! Is SO!

9. INSCRIBED ANGLE THEOREM Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. x x

10. INSCRIBED ANGLE THEOREM Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.

11. INSCRIBED ANGLE THEOREM Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Inscribed Angle Y 110 55 Z Intercepted Arc

12. Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Find the value of x and y in the figure. • X = 20° • Y = 60° P 40 Q 50 y S x R T

13. Corollary 1. If two inscribed angles intercept the same arc, then the angles are congruent.. Find the value of x and yin the figure. • X = 50° • Y = 50° P Q y 50 S R x T

14. An angle formed by a chord and a tangent can be considered an inscribed angle. 2x

15. An angle formed by a chord and a tangent can be considered an inscribed angle. P Q S R mPRQ = ½ mPR

16. What is mPRQ ? P Q 60 S R

17. An angle inscribed in a semicircle is a right angle. P 180 R

18. An angle inscribed in a semicircle is a right angle. P 180 90 S R

19. Interior Angles • Angles that are formed by two intersecting chords. (Vertex IN the circle) A D B C

20. Interior Angle Theorem The measure of the angle formed by the two chords is equal to ½ the sum of the measures of the intercepted arcs.

21. A D 1 B C Interior Angle Theorem The measure of the angle formed by the two chords is equal to ½ the sum of the measures of the intercepted arcs.

22. Interior Angle Theorem 91 A C x° y° B D 85

23. Exterior Angles • An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle. (vertex OUT of the circle.)

24. Exterior Angles • An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle. j k k 1 j j k 1 1

25. k k j k j j 1 1 3 Exterior Angle Theorem • The measure of the angle formed is equal to ½ the difference of the intercepted arcs.

26. Find • <C = ½(265-95) • <C = ½(170) • m<C = 85°

27. D 6 C E Q 5 3 A F 2 1 4 G PUTTING IT TOGETHER! • AF is a diameter. • mAG=100 • mCE=30 • mEF=25 • Find the measure of all numbered angles.

28. Inscribed Quadrilaterals • If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. P Q mPSR + mPQR = 180  S R