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CIRCLES 2 Moody Mathematics

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ANGLE PROPERTIES: Moody Mathematics Let’s review the methods for finding the arcs and the different kinds of angles found in circles.

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Moody Mathematics The measure of a minor arc is the same as… …the measure of its central angle.

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Example:

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Moody Mathematics The measure of an inscribed angle is… …half the measure of its intercepted angle.

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Example:

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Moody Mathematics The measure of an angle formed by a tangent and secant is … …half the measure of its intercepted arc.

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Example:

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Moody Mathematics The measure of one of the vertical angles formed by 2 intersecting chords...is half the sum of the two intercepted arcs.

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Example:

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Moody Mathematics The measure of an angle formed by 2 secants intersecting outside of a circle is… …half the difference of the measures of its two intercepted arcs.

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Example:

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Moody Mathematics The measure of an angle formed by 2 tangents intersecting outside of a circle is… …half the difference of the measures of its two intercepted arcs.

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Example:

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PROPERTIES: Complete the theorem relating the objects pictured in each frame. PROPERTIES: Complete the theorem relating the objects pictured in each frame. Moody Mathematics

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Note: Many of our theorems begin the same way, “In the same circle, or in congruent circles…” Note: Many of our theorems begin the same way, “In the same circle, or in congruent circles…” Moody Mathematics

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So: We will just start “In the same circle*…” where the * represents the rest of the phrase. So: We will just start “In the same circle*…” where the * represents the rest of the phrase. Moody Mathematics

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All radii in the same circle,* …...are congruent.

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In the same circle,* Congruent central angles......intercept congruent arcs.

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In the same circle,* Congruent Chords......intercept congruent arcs.

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Tangent segments from an exterior point to a circle…...are congruent.

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The radius drawn to a tangent at the point of tangency…...is perpendicular to the tangent.

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If a diameter (or radius) is perpendicular to a chord, then…...it bisects the chord… …and the arcs.

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In the same circle,* Congruent Chords......are equidistant from the center.

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Example: Given a circle of radius 5” and two 8” chords. Find their distance to the center.

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Moody Mathematics If two Inscribed angles intercept the same arc......then they are congruent.

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If an inscribed angle intercepts or is inscribed in a semicircle …...then it is a right angle.

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If a quadrilateral is inscribed in a circle then each pair of opposite angles …...must be supplementary. (total 180 o )

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If 2 chords intersect in a circle, the lengths of segments formed have the following relationship:

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Example:

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Moody Mathematics If 2 secants intersect outside of a circle, their lengths are related by…

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Example:

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Moody Mathematics If a secant and tangent intersect outside of a circle, their lengths are related by…

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Example:

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Let’s Practice!

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Moody Mathematics Example: Given

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Moody Mathematics Example:

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Moody Mathematics Example:

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Moody Mathematics Example:

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Moody Mathematics Example: Given a circle of radius 13” and two 24” chords. Find their distance to the center.

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Moody Mathematics Example:

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Moody Mathematics Example:

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Moody Mathematics Example:

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Moody Mathematics Example:

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Moody Mathematics Example:

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Moody Mathematics Example:

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Example: Of the following quadrilaterals, which can not always be inscribed in a circle? A.Rectangle B.Rhombus C.Square D.Isosceles Trapezoid

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Moody Mathematics Example:

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Moody Mathematics Example:

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Moody Mathematics Example: Regular Hexagon ABCDEF is inscribed in a circle.

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THE END! Now go practice!

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