Obj. 58 Angle Relationships

The student is able to (I can):

• Find the measures of angles formed by lines that intersect circles

• Use angle measures to solve problems

If a tangent and a secant (or a chord) intersect at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc.

F

L•Y

���LF is a secant.���LY is a tangent.

�∠ =1

m FLY mFL2

Example Find each measure:

1. m∠EFH

2.

180 — 122 = 58º

�mGF

∠ = = °1

m EFH (130) 652

58º

� = = °mGF 2(58) 116

If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the intercepted arcs.

1111G

R

A

D

� �( )∠ = +1

m 1 mDG mRA2

Example Find each measure.

1. m∠1

2. m∠2

m∠2 = 180 — m∠1

= 180 — 80 = 100º

99º

61º

12

( )∠ = +1

m 1 99 612

= 80º

If secants or tangents intersect outside a circle, the measure of the angle formed is half the difference between the intercepted arcs.

M O N

E

Y

1

� �( )∠ = −1

m 1 mNY mOE2

Example Find each measure

1. m∠K

2. x

186º62º

K

26º

94º

∠ = −1

m K (186 62)2

= 62º

= −1

26 (94 x)2 xº

52 = 94 — x

x = 42º