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Geometry 10.4 Other Angle Relationships in Circles

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles2 Goal Use angles formed by tangents, secants, and chords to solve problems.

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles3 Review Note: in solving an equation with fractions, one of the first things to do is always clear the fractions.

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles4 You do it. Solve:

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles5 Review The measure of an inscribed angle is equal to one-half the measure of the intercepted arc What if one side of the angle is tangent to the circle?

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles6 Theorem 10.2: Tangent-Chord A B C 12 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.

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles7 Simplified Formula a b 1 2

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles8 Example 1 A B C

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles9 Example 2. Solve for x. A B C 4x (10x – 60)

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles10 If two lines intersect a circle, where can the lines intersect each other? On the circle. Inside the circle. Outside the circle. We already know how to do this.

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles11 Theorem (Inside the circle) A B C D 1 If two chords intersect in a circle, then the measure of the angle is one-half the sum of the intercepted arcs.

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles12 Simplified Formula 1 a b

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles13 Example 3Find m1. A B C D

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles14 Example 4Solve for x. A B C D x 100 Check: = ÷ 2 = 60

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles15 Your turn. Solve for x & y. A B C D x M y K P O 20 32

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles16

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles17 Secant-Secant C A B D 1

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles18 Simplified Formula 1b a

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles19 Secant-Tangent C A B 1

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles20 Simplified Formula a b 1

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles21 Tangent-Tangent A B 1 C

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles22 Simplified Formula 1 a b

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles23 Intersection Outside the Circle Secant-SecantSecant-TangentTangent-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.)

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles24 Example 5Find m

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles25 Example 6Find m

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles26 Example 7Find m ? 360 – 210 = 150 k m Rays k and m are tangent to the circle.

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles27 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.

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles28 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.

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles29 Answers [2 105] 3.60 [½( )] 4.65 [ ½(190 – 60)] 5.90 [ ½(270 – 90)] [ ½(80 – 35)] 7.88 [ ½( )]

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Thursday, March 23, 2:46 Geometry 10.4 Other Angle Relationships in Circles30 Homework

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