Geometry – Inscribed and Other Angles

1. Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A C C A B B C B All three of these inscribed angles intercept arc AB.

2. Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A C C A B B C B All three of these inscribed angles intercept arc AB. Theorem : An inscribed angle is equal to half of its intercepted arc.

3. Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A 32° C 1 C 200° A 2 B 40° 3 B C B Theorem : An inscribed angle is equal to half of its intercepted arc. EXAMPLE : Find the measure of angles 1 , 2 and 3.

4. Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A 32° C 1 C 200° A 2 B 40° 3 B C B Theorem : An inscribed angle is equal to half of its intercepted arc. EXAMPLE : Find the measure of angles 1 , 2 and 3.

5. Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A 32° C 1 C 200° A 2 B 40° 3 B C B Theorem : An inscribed angle is equal to half of its intercepted arc. EXAMPLE : Find the measure of angles 1 , 2 and 3.

6. Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A 32° C 1 C 200° A 2 B 40° 3 B C B Theorem : An inscribed angle is equal to half of its intercepted arc. EXAMPLE : Find the measure of angles 1 , 2 and 3.

7. Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A ? C 86° C ? A 25° B 18° ? B C B Theorem : An inscribed angle is equal to half of its intercepted arc. EXAMPLE #2 : Find the measure of arc AB in each example.

8. Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A ? C 86° C ? A 25° B 18° ? B C B Theorem : An inscribed angle is equal to half of its intercepted arc. EXAMPLE #2 : Find the measure of arc AB in each example. Take notice that the arc is two time bigger than the angle.

9. Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A ? C 86° C ? A 25° B 18° ? B C B Theorem : An inscribed angle is equal to half of its intercepted arc. EXAMPLE #2 : Find the measure of arc AB in each example. Take notice that the arc is two times bigger than the angle.

10. Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A 36° C 86° C ? A 25° B 18° 50° B C B Theorem : An inscribed angle is equal to half of its intercepted arc. EXAMPLE #2 : Find the measure of arc AB in each example. Take notice that the arc is two times bigger than the angle.

11. Geometry – Inscribed and Other Angles Theorem : An angle formed by a tangent line and a chord is equal to half of its intercepted arc. A 1 B

12. Geometry – Inscribed and Other Angles Theorem : An angle formed by a tangent line and a chord is equal to half of its intercepted arc. A 1 B EXAMPLE : If arc AB = 65°, find the measure of angle 1.

13. Geometry – Inscribed and Other Angles Theorem : An angle formed by a tangent line and a chord is equal to half of its intercepted arc. A 1 B EXAMPLE : If arc AB = 65°, find the measure of angle 1.

14. Geometry – Inscribed and Other Angles Theorem : An angle formed by a tangent line and a chord is equal to half of its intercepted arc. X A 1 B EXAMPLE #2 : If arc AXB = 300°, find the measure of angle 1.

15. Geometry – Inscribed and Other Angles Theorem : An angle formed by a tangent line and a chord is equal to half of its intercepted arc. X A 1 B EXAMPLE #2 : If arc AXB = 300°, find the measure of angle 1.

16. Geometry – Inscribed and Other Angles Theorem : An angle formed by two chords is equal to half of the sum of the intercepted arcs C A X B D

17. Geometry – Inscribed and Other Angles Theorem : An angle formed by two chords is equal to half of the sum of the intercepted arcs C 40° A X B D 42° EXAMPLE : Arc AC = 40° and arc BD = 42°. Find the measure of angle CXA.

18. Geometry – Inscribed and Other Angles Theorem : An angle formed by two chords is equal to half of the sum of the intercepted arcs C ? A X B D 50° EXAMPLE # 2 : Angle CXA = 40° and arc BD = 50°. Find the measure of arc CA.

19. Geometry – Inscribed and Other Angles Theorem : An angle formed by two chords is equal to half of the sum of the intercepted arcs C y A X B D 50° EXAMPLE # 2 : Angle CXA = 40° and arc BD = 50°. Find the measure of arc CA.

20. Geometry – Inscribed and Other Angles Theorem : An angle formed by two secants is equal to half of the difference of the intercepted arcs. ( a secant is a line that cuts through a circle ) D C X A B

21. Geometry – Inscribed and Other Angles Theorem : An angle formed by two secants is equal to half of the difference of the intercepted arcs. ( a secant is a line that cuts through a circle ) D C 75° 23° X A B EXAMPLE : Arc BD = 75° and arc CA = 23°. Find the measure of angle “x” .

22. Geometry – Inscribed and Other Angles Theorem : An angle formed by two secants is equal to half of the difference of the intercepted arcs. ( a secant is a line that cuts through a circle ) D C 75° 23° X A B EXAMPLE : Arc BD = 75° and arc CA = 23°. Find the measure of angle “x” .