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10.6 Secants, Tangents, &amp; Angle Measures

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10.6 Secants, Tangents, &amp; Angle Measures. Objectives. Find measures of angles formed by lines intersecting on, inside, or outside a circle. A. D. O. C. B. Secants and Interior Angles.

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Objectives
• Find measures of angles formed by lines intersecting on, inside, or outside a circle.

A

D

O

C

B

Secants and Interior Angles

Theorem 10.12:If two secants intersect in the interior of a , then the measure of an  formed is ½ the measure of the sum of the arcs intercepted by the secants that created the .

mAOC = ½ (m arc AC + m arc DB)

mAOD = ½ (m arc AD + m arc CB)

Example 1:

Method 2

Answer: 98

D

A

O

C

B

F

Secants, Tangents and Angles

Theorem 10.13:If a secant and a tangent intersect at the point of tangency, then the measure of each  formed is ½ the measure of its intercepted arc.

mDOB = ½ (m arc OB) mCOB = ½ (m arc OFB)

A

G

B

F

H

O

I

C

E

D

Secants – Tangents & Exterior Angles

Theorem 10.14:If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the  formed is ½ the measures of the difference of the intercepted arcs.

mBOC = ½ (m arc BC – m arc HI) mAOC= ½ (m arc AC – m arc AI)mAED= ½ (m arc ABD – m arc AD)

Example 3:

Find x.

Theorem 10.14

Multiply each side by 2.

Add x to each side.

Subtract 124 from each side.

Answer: 17

Your Turn:

Find x.

Answer: 111

Example 4:

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 one-half the difference of the measure of the two intercepted arcs.

Example 4:

Multiply each side by 2 and simplify.

Add 360 to each side.

Divide each side by 2.

Answer: 220

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

Your Turn:

Answer: 75

Example 5:

Find x.

Multiply each side by 2.

Add 40 to each side.

Divide each side by 6.

Answer: 25

Your Turn:

Find x.

Answer: 9

Assignment
• Pre-AP GeometryPg. 564 #12 - 32
• Geometry:Pg. 564 #3 – 7, 12 - 26