Transparency 10-6. 5-Minute Check on Lesson 10-5. Determine whether each segment is tangent to the given circle . BC 2. QR Find x. Assume that the segments that appear to be tangents are tangents.

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5-Minute Check on Lesson 10-5

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Determine whether each segment is tangent to the given circle.

BC 2. QR

Find x. Assume that the segments that appear to be tangents are tangents.

3. 4.

5. What is the measure of PS?

Yes

No

x = 20√5

or 44.72

x = 12

Standardized Test Practice:

A

B

C

D

18

10

B

12

14

Click the mouse button or press the Space Bar to display the answers.

Lesson 10-6

Secants, Tangents, and Angle Measures

Objectives

Find measures of angles formed by lines intersecting on or inside a circle

Find measures of angles formed by lines intersecting outside a circle

Vocabulary

Secant – a line that intersects a circle in exactly two points

y

x

Circles – Secants

Interior Angles formed by a Secant & a Tangentm3 = ½ (m Arc JMK)

m4 = ½ (m Arc JLK)

M

3

K

2

4

1

J

P

Center

Interior Angles formed by 2 Secantsm1 = ½ (m Arc MJ + m Arc LK)

m2 = ½ (m Arc MK + m Arc JL)

L

Circles – External Angles

Two Secants

Secant & Tangent

Two Tangents

J

J

J

K

K

L

S

T

T

M

M

M

N

mJ = ½(m Arc TM – m Arc TK)

mJ = ½|m Arc TM – m Arc TK|

mJ = ½(m Arc TMS – m Arc TS)

mJ = ½|m Arc TMS – m Arc TS|

mJ = ½(m Arc MN – m Arc LK)

mJ = ½|m Arc MN – m Arc LK|

Find if and

Example 6-1a

Method 1

Example 6-1b

Method 2

Answer: 98

Find if and

Example 6-1d

Answer: 138

Find if and

Example 6-2a

Answer: 55

Find if and

Example 6-2c

Answer: 58

Example 6-3a

Find x.

Theorem 10.14

Multiply each side by 2.

Add x to each side.

Subtract 124 from each side.

Answer: 17

Example 6-3c

Find x.

Answer: 111

Example 6-4a

JEWELRY A jeweler wants to craft a pendant with the shape shown. Use the figure to determine the measure of the arc at the bottom of the pendant.

Let x represent the measure of the arc at the bottom of the pendant. Then the arc at the top of the circle will be 360 – x. The measure of the angle marked 40° is equal to 1/2 the difference of the measure of the two intercepted arcs.

Multiply each side by 2 and simplify.

Add 360 to each side.

Divide each side by 2.

Answer: 220

PARKS Two sides of a fence to be built around a circular garden in a park are shown. Use the figure to determine the measure of

Example 6-4c

Answer: 75

Example 6-5a

Find x.

Multiply each side by 2.

Add 40 to each side.

Divide each side by 6.

Answer: 25

Example 6-5c

Find x.

Answer: 9

Summary & Homework

Summary:

The measure of an angle formed by two secant lines is half the positive difference of its intercepted arcs

The measure of angle formed by a secant and tangent line is half its intercepted arc