1. Aim: How do we find angle measurements inside and outside of circles? Do now: A security camera located at point S views part of a circular area by rotating 30 o. How many degrees of the circle’s circumference does the camera observe? 60 o S 30 o

2. Basic angles 1. Central angles2. Inscribed angles ab a b a = ½ ba = b An angle inscribed in a semicircle (diameter) is a right angle. a = 70 o, then b = 70 o If… a = 36 o, then b = 36 o b = 170 o, then a = 170 o a = 70 o, then b = 140 o If… a = 36 o, then b = 72 o b = 170 o, then a = 85 o

3. Special angles Group 1: Tangent - Chord a x x = ½ a Mnemonic a x Since a = 180 o, x = 90 o. x = ½ a. a = 80 o, then x = 40 o If… a = 228 o, then x = 114 o x = 25 o, then a = 50 o

4. Special angles Group 2: Secant - Secant x = ½ (a - b) a x a = 100 o and b = 60 o, then x = 20 o If… b = 60 o and x = 25 o, then a = 110 o a = 70 o and x = 10 o, then b = 50 o Tangent - SecantTangent - Tangent x xa b b b a Mnemonic

5. Special angles Group 3: Chord - Chord a x = ½ (a + b) Mnemonic The three formulas: x = ½ a x = ½ (a-b) x = ½ (a+b) b x a = 90 o and b = 10 o, then x = 50 o If… b = 100 o and x = 70 o, then a = 40 o a = 200 o and x = 150 o, then b = 100 o Review

6. Putting it all together Given: In circle O, angle AOB = 60 o, angle BEA = 80 o, and BOD is a diameter. Find: Arc AB Angle AOD Arc AD Arc CD Arc BC Angle BCA Angle BEC Angle CBE 60 o 30 o O B A C D 80 o 100 o 80 o 120 o 100 o 50 o E

7. Regents problems 96 o 48 o

8. Regents problems 180 o 9 = 20 o AC: 20 o x 7 = 140 o BC: 20 o x 2 = 40 o x = ½ (a - b) x = ½ (140 o - 40 o ) x = ½ (100 o ) x = 50 o m angle CPA = 50 o m arc ACB = 180 o a x b