warm-up

Warm-Up

A circle and an angle are drawn in the same plane. Find all possible ways in which the circle and angle intersect at two points.

9.4 – Angles Formed by Secants and Tangents

Topics:

Theorems about measures of arcs intercepted by angles

Classification of Angles with Circles

Case 1: Vertex is on the circle

Two secants Secant and tangent

Classification of Angles with Circles

Case 2: Vertex is inside the circle

Two secants

Classification of Angles with Circles

Case 3: Vertex is outside the circle

Two secants Secant and tangent

Two tangents

Vertex on Circle – Secant and Tangent

Secant and tangent

Vertex on Circle – Secant and Tangent

A

C

m AVC

Part 1 – The secant-tangent angle is a right angle.

90 mAV 180

V

P

Vertex on Circle – Secant and Tangent

V

P

C

Part 2 – The secant-tangent angle is acute.

A1

2

3

Copy and complete the table.

mAV m 1 m 2 m PVC m AVC

120 120 30 60

100

80

x

Vertex on Circle – Secant and Tangent

V

P

C

Part 3 – The secant-tangent angle is obtuse.

A1

2

3Copy and complete the table.

mAXV m 1 m 2 m PVC m AVC

200 160 10 100

220

240

x

X

Theorem

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

Vertex Inside Circle – Two Secants

VP

C

Find the relationship between and the measures of

A12

Copy and complete the table.

m AVC 1 and 2.

D

B

mAC mBD m 1 m 2 m AVC

160 40 80 100

180

2001x

m DVB20 100

70

602x

Theorem

The measure of an angle formed by two secants or chords that intersect in the interior of a circle is _______ the _______ of the measures of the arcs intercepted by the angle and its vertical angle.

one-half sum

Vertex Outside Circle – Two Secants

V C

Find the relationship between and the measures of A

1

2

Copy and complete the table.

mBD mAC m 1 m 2 m AVC

200 40 100 20

250

1001x

D

m 1 2 and AVC.

2x

80

60

50

B

Theorem

The measure of an angle formed by two secants that intersect in the exterior of a circle is ________ the _________ of the measures of the intercepted arcs.

one-half difference

Example 1

Find mDAB, mCBA, andmDGC.

Given: AB is tangent to circle Pat A, mAC = 60, mCD = 70,and mDE = 80

mDAB.

Because the vertex of DAB is ON the circle,

1

mAD2

1

60 702

65

Example 1

Find mDAB, mCBA, andmDGC.

Given: AB is tangent to circle Pat A, mAC = 60, mCD = 70,and mDE = 80

The vertex of CBA is outside the circle, so

mCBA 1

mEA mAC2

1

150 602

45

Example 1

Find mDAB, mCBA, andmDGC.

Given: AB is tangent to circle Pat A, mAC = 60, mCD = 70,and mDE = 80

The vertex of DGC is inside the circle, so

mDGC 1

mEA mCD2

1

150 702

110

Practice Problem

Find mWXY, m1, andm2.

Given: XY is tangent to circle Tat X, mUV = 90, mVX = 130,and mUW = 20 V

X

T

W Z

Y

1

2

Homework

p.593 #10-26