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 30o. How many degrees of the circle’s circumference does the camera observe?

60o

60o

S

30o

Basic angles

1. Central angles 2. Inscribed angles

a ba

b

a = ½ ba = b

An angle inscribed in a semicircle(diameter) is a right angle.

a = 70o, then b = 70o

If…

a = 36o, then b = 36o

b = 170o, then a = 170o

a = 70o, then b = 140o

If…

a = 36o, then b = 72o

b = 170o, then a = 85o

Special anglesGroup 1:

Tangent - Chord

ax

x = ½ a

Mnemonic

a

x

Since a = 180o, x = 90o.

x = ½ a.a = 80o, then x = 40o

If…

a = 228o, then x = 114o

x = 25o, then a = 50o

Special anglesGroup 2:

Secant - Secant

x = ½ (a - b)

a x

a = 100o and b = 60o, then x = 20o

If…

b = 60o and x = 25o, then a = 110o

a = 70o and x = 10o, then b = 50o

Tangent - SecantTangent - Tangent

x xab bba

Mnemonic

Special anglesGroup 3:

Chord - Chord

a

x = ½ (a + b)

Mnemonic

The three formulas:

x = ½ a

x = ½ (a-b)

x = ½ (a+b)

bx

a = 90o and b = 10o, then x = 50o

If…

b = 100o and x = 70o, then a = 40 o

a = 200o and x = 150o, then b = 100o

Review

Putting it all togetherGiven: In circle O,

angle AOB = 60o, angle BEA = 80o, and BOD is a diameter.

Find:

Arc AB

Angle AOD

Arc AD

Arc CD

Arc BC

Angle BCA

Angle BEC

Angle CBE

60o

60o

30o

O

B

A

CD

80o

100o

80o120o

120o

100o

50o

E

Regents problems

96o

96o

48o

Regents problems

180o

9= 20o

AC: 20o x 7 = 140o

BC: 20o x 2 = 40o

x = ½ (a - b)

x = ½ (140o - 40o)

x = ½ (100o)

x = 50o

m angle CPA = 50o

m arc ACB = 180o

a

xb