10.5.1 Apply Other Angle Relationships in Circles

If a chord intersects a tangent then the measure of the angle is one half the measure of the intercepted arc

A

B

C

1 2

m1= ½ mACB

m2= ½ mAB

The measure of each angle is one half the sum of the intercepted arcs

A

BC1

2

D

3

4

Since 1 2

m3 = m4 = ½ (mCA + mBD)

m1 = m2 = ½ (mCD + m AB)

If an angle is outside the circle the measure of the circle is one half the difference of the intercepted arc 3 cases, same rule:

A

B

C

1D

A

C

1

A

B

C

1

B

m1 = ½ (mAB – mCD)

m1 = ½ (mABC – mCA)

m1 = ½ (mAC – mCB)

A

B

C

D

E

F

60⁰

80⁰

40⁰

)(2

160 mABmACB

mABmACB 120

mABmACB 360

mACB*2480

mACB240

mAB120

)(2

180 mBCmCAB

mBCmCAB 160

mBCmCAB 360

mCAB*2520mCAB260

mBC100

)(2

140 mACmABC

mACmABC 80mACmABC 360

mABC*2440mABC220mAC140

p. 683   1 – 6, 10 – 13, 16 - 20, 23 – 27odd, 32 -

38even