5-Minute Check on Lesson 10-55-Minute Check on Lesson 10-55-Minute Check on Lesson 10-55-Minute Check on Lesson 10-5 Transparency 10-6

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

Determine whether each segment is tangent to the given circle.

1. 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?Standardized Test Practice:

A CB D10 12 14 18

Yes No

x = 12x = 20√5or 44.72

B

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

Interior Angles formed by a Secant & a Tangent

m3 = ½ (m Arc JMK)m4 = ½ (m Arc JLK)

Circles – Secants

Interior Angles formed by 2 Secants

m1 = ½ (m Arc MJ + m Arc LK)m2 = ½ (m Arc MK + m Arc JL)

Center

J

K

L

M

12

P

3

4

Circles – External Angles

JJ

K KL

T

MMN

J

ST

M

mJ = ½(m Arc MN – m Arc LK)mJ = ½|m Arc MN – m Arc LK|

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|

Two Secants Two TangentsSecant & Tangent

Find if and

Method 1

Method 2

Answer: 98

Answer: 138

Find if and

Find if and

Answer: 55

Answer: 58

Find if and

Find x.

Theorem 10.14

Multiply each side by 2.Add x to each side.Subtract 124 from each side.

Answer: 17

Find x.

Answer: 111

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

Answer: 75

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

Find x.

Multiply each side by 2.

Add 40 to each side.

Divide each side by 6.

Answer: 25

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

• Homework: – pg 564-566; 12-14, 18-20; 23-24, 26, 29, 34-36