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**10.4 Other Angle Relationships in Circles**

Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Goal Use angles formed by tangents, secants, and chords to solve problems. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Review Note: in solving an equation with fractions, one of the first things to do is always “clear the fractions”. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

You do it. Solve: Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Review The measure of an inscribed angle is equal to one-half the measure of the intercepted arc. 40 80 What if one side of the angle is tangent to the circle? Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Theorem 10.2: Tangent-Chord**

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. B C 2 1 A Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Simplified Formula a b 1 2 Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Example 1 B C 160 200 80 A Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Example 2. Solve for x. B C (10x – 60) 4x A Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

If two lines intersect a circle, where can the lines intersect each other? Inside the circle. We already know how to do this. On the circle. Outside the circle. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Theorem 10.13 (Inside the circle)**

If two chords intersect in a circle, then the measure of the angle is one-half the sum of the intercepted arcs. A C 1 B D Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Simplified Formula 1 a b Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Example 3 Find m1. A C 30 1 B 80 D Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Example 4 Solve for x. A C 20 60 B x 100 Check: = 120 120 ÷ 2 = 60 D Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Your turn. Solve for x & y. M y K P O 20 32 A B C D x 75 85 Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Intersection Outside the Circle Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Secant-Secant A C 1 D B Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Simplified Formula 1 b a Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Secant-Tangent A C 1 B Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Simplified Formula a b 1 Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Tangent-Tangent A C 1 B Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Simplified Formula 1 a b Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Intersection Outside the Circle**

Secant-Secant Secant-Tangent Tangent-Tangent In all cases, the measure of the exterior angle is found the same way: One-half the difference of the larger and smaller arcs. (Click the titles above for Sketchpad Demonstrations.) Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Example 5 Find m1. 80 35 1 10 Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Example 6 Find m1. 120 70 1 25 Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Example 7 Find m1. k Rays k and m are tangent to the circle. 210 ? 150 1 30 m 360 – 210 = 150 Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

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. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

What can you do? Open a book to page 624. Do problems 2 – 7. Carefully consider what the situation is and use the correct formula. 6 minutes. Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Answers 210 [2 105] 60 [½( )] 65 [ ½(190 – 60)] 90 [ ½(270 – 90)] 22.5 [ ½(80 – 35)] 88 [ ½( )] Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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**Geometry 10.4 Other Angle Relationships in Circles**

Homework Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles

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10.4 Other Angles Relationships In Circles. Theorem A tangent and a chord intersect at a point, it makes angles that are ½ the intercepted arc.

10.4 Other Angles Relationships In Circles. Theorem A tangent and a chord intersect at a point, it makes angles that are ½ the intercepted arc.

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