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A circle is the set of all points that are a given distance from a fixed point. The given distance is called the radius. The fixed point is called the center. 55

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A circle is the set of all points that are a given distance from a fixed point. The given distance is called the radius. The fixed point is called the center. Here are two points. Which one is part of the circle?

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A circle is the set of all points that are a given distance from a fixed point. The given distance is called the radius. The fixed point is called the center. Secant is a line that intersects Chord is a segment that joins a circle in two points. two points on a circle. Diameter is a chord that passes through the center.

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Definition: A tangent is a line that meets a circle in exactly one point, called the point of tangency.

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Theorem: If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. Definition: A tangent is a line that meets a circle in exactly one point, called the point of tangency. l Q Proof: Suppose AP is not perpendicular to tangent line l.

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A B C P In the diagram, segments PB and PC are tangent to circle A. Prove that PB and PC are congruent. Theorem: The tangent segments drawn to a circle from an external point are congruent.

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An arc is part of a circle. An arc is the set of points consisting of 2 points on a circle and all the points on the circle between them. FHG

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Central angle of a circle is an angle whose vertex is the center of the circle. The measure of a minor arc is defined to be the measure of its central angle. The measure of a circle is 360 Theorem: In the same circle or in congruent circles. (1) Congruent central angles have congruent arcs (2) Congruent arcs have congruent central angles (3) Congruent chords have congruent arcs (4) Congruent arcs have congruent chords. An arc is part of a circle. An arc is the set of points consisting of 2 points on a circle and all the points on the circle between them. FHG m EFG = mEG

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Theorem: A diameter that is perpendicular to a chord, bisects the chord and its arcs.

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Any “2x2 converse” is also true. _______ ___________________ 1 2 3 4

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An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. Theorem: The measure of an inscribed angle is equal to half the measure of its intercepted arc. BAD = ½ (BD)

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Theorem: The measure of an inscribed angle is equal to half the measure of its intercepted arc. An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. Theorem: If two inscribed angles intercept the same arc, then they are congruent. Theorem: An angle inscribed in a semicircle is a right angle. In the diagram, BAC BDC because they are both inscribed angles and they intercept the same arc. BAD = ½ (BD)

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Theorem: An angle formed by two chords intersecting inside a circle is equal to half the sum of the intercepted arcs. ARE = ½ (AE + DC)

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Theorem: An angle formed by two chords intersecting inside a circle is equal to half the sum of the intercepted arcs. ARE = ½ (AE + DC) Find the measure of 1. AED 2. CAE 3. BFA F 65 60 55

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APE = ½ (AE – CB) Theorem: An angle formed by two secants, two tangents, or by a secant and a tangent drawn from a point outside a circle is equal to half the difference of the intercepted arcs.

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APE = ½ (AE – CB) APE = ½ (AE – AC) APE = ½ (ANE – AE) Theorem: An angle formed by two secants, two tangents, or by a secant and a tangent drawn from a point outside a circle is equal to half the difference of the intercepted arcs.

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Find the values of x and y. x = 31 y = 46.5

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Theorem: The measure of an angle formed by a tangent and a chord of a circle is half the measure of the arc between them. ABC = ½ AB

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Triangle ABC is an isosceles triangle with AB = AC. Find the measures of x, y, and z. x = 140 y = 70 z = 40 A C B

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is tangent to circle P; is a diameter; AB = 30 , CD = 40 , DE =50 Find the measure of each numbered angle. 1 = 40 30 40 50 110 130 2 = 15 3 = 25 4 = 40 5 = 65 6 = 100 7 = 45 8 = 30 9 = 25 10 = 65

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B A E D C In the diagram, radius AB is perpendicular to radius AC. Point E is chosen randomly on minor arc BC, and CD is constructed perpendicular to ray BE at point D. Using Geometer’s Sketchpad, make a conjecture about the relationship between the lengths of segments CD and DE and prove you are correct.

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