Geometry 10.4 Other Angle Relationships in Circles.

Geometry

10.4 Other Angle Relationships in Circles

Thursday, March 23, 2:46

Geometry 10.4 Other Angle Relationships in Circles 2

Goal

Use angles formed by tangents, secants, and chords to solve problems.



Review

115 4 9

21

2 15 2 4 92

30 4 9

39 4

399.75

4

x

x

x

x

x

Note: in solving an equation with fractions, one of the first things to do is always “clear the fractions”.



You do it. Solve: 120 4 10

4x

120 4 10

41

4 20 4 4 104

80 4 10

90 4

22.5

x

x

x

x

x



Review

The measure of an inscribed angle is equal to one-half the measure of the intercepted arc.

8040

What if one side of the angle is tangent to the circle?



Theorem 10.2: Tangent-Chord

A

BC

12

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of the intercepted arc.

1 122 and1 2m mA m m CAB B



Simplified Formula

ab

12

12

12

1

2

m a

m b



Example 1

Find the and .mAB mBCA

1280

160

mAB

mAB

A

BC

80

360 160

200

mBCA

160200



Example 2. Solve for x.

A

BC

4x

(10x – 60)

124 (10 60)

8 10 60

2 60

30

x x

x x

x

x



If two lines intersect a circle, where can the lines intersect each other?

On the circle.

Inside the circle.

Outside the circle.

We already know how to do this.



Theorem 10.13 (Inside the circle)

A

B

C

D

1

If two chords intersect in a circle, then the measure of the angle is one-half the sum of the intercepted arcs.

121m mAB mCD



Simplified Formula

1a

b

121m a b



Example 3 Find m1.

A

B

C

D

130

80 1

2

12

1 30 80

(110)

55

m



Example 4 Solve for x.

A

B

C

D

6020

x

1260 20

120 20

100

x

x

x

100

Check:

100 + 20 = 120

120 ÷ 2 = 60



Your turn. Solve for x & y.

A

B

C

D

x75 85M

y

K

P

O

20

32

132 20

264 20

44

y

y

y

175 85

280

x

x





Secant-Secant

C

A

BD

121m mAB mCD

1



Simplified Formula

1 b a

121m a b



Secant-Tangent

C

A

B

121m mAB mBC

1



Simplified Formula

a

b1

121m a b



Tangent-Tangent

A

B

121m mACB mAB

1

C



Simplified Formula

1 ab

121m a b



Example 5 Find m1.

1 8010

1 12 21 80 10 (70) 35m

35



Example 6 Find m1.

1

12070

1 12 21 120 70 (50) 25m

25



Example 7 Find m1.

1

210

150

1 12 21 210 150 (60) 30m

30?

360 – 210 = 150

k

m

Rays k and m are tangent to the circle.



How to remember this: If the angle vertex is on the circle, its

measure is one-half the intercepted arc.

If an angle vertex is inside the circle, its measure if one-half the sum of the intercepted arcs.

If an angle vertex is outside the circle, its measure is one-half the difference of the intercepted arcs.

Geometry 10.4 Other Angle Relationships in Circles.

Documents

angle relationships

angle vertex

inscribed angle

intercepted arcs

circlesreviewthe measure

circlesgoaluse angles