Angle Relationships

The student is able to (I can):

• Find the measure of an inscribed angle

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

• Use angle measures to solve problems

inscribed angle An angle whose vertex is on the circle and whose sides contain chords of the circle.

The measure of an inscribed angle is ½ the measure of its intercepted arc.

A

�∠ =1

m AHR AR

•H

I

R

�∠ =1

m AHR AR2

� = ⋅ ∠AR 2 m AHR

Example Find each measure:

1. m∠MAPM

A

P

110º�( )∠ =1

m MAP mMP2

= = °1(110) 55

2

2.

= 2(24)

= 48º

J

Y

O

24º24º24º24º

�mJY� = ∠mJY 2(m JOY)

If inscribed angles intercept the same arc, then the angles are congruent.

R

E∠RED ≅ ∠RAD

A D

An inscribed angle intercepts a semicircle if and only if it is a right angle.

•

If a quadrilateral is inscribed in a circle, its opposite angles are supplementary.

F

R

E

FRED is inscribed in the circle.

D

m∠F + m∠E = 180ºm∠R + m∠D = 180º

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

���LF is a secant.���LY is a tangent.

L•Y

�∠ =1

m FLY mFL2

Example Find each measure:

1. m∠EFH

2. �mGF

∠ = = °1

m EFH (130) 652

58º2.

180 — 122 = 58º

�mGF

� = = °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

� �( )∠ = +1

m 1 mDG mRA2

A

D

Example Find each measure.

1. m∠1

99º

61º

12

( )∠ = +1

m 1 99 612

= 80º

2. m∠2

m∠2 = 180 — m∠1

= 180 — 80 = 100º

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

1

E

Y� �( )∠ = −

1m 1 mNY mOE

2

Example Find each measure

1. m∠K

2. x

186º62º

K

26º

∠ = −1

m K (186 62)2

= 62º

2. x 26º

94º

= −1

26 (94 x)2 xº

52 = 94 — x

x = 42º