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CIRCLES 2 Moody Mathematics

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

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**The measure of a minor arc is the same as…**

…the measure of its central angle. Moody Mathematics

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Example: Moody Mathematics

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**The measure of an inscribed angle is…**

…half the measure of its intercepted angle. Moody Mathematics

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Example: Moody Mathematics

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**The measure of an angle formed by a tangent and secant is …**

…half the measure of its intercepted arc. Moody Mathematics

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Example: Moody Mathematics

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**...is half the sum of the two intercepted arcs.**

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

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Example: Moody Mathematics

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**…half the difference of the measures of its two intercepted arcs.**

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. Moody Mathematics

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Example: Moody Mathematics

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**…half the difference of the measures of its two intercepted arcs.**

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. Moody Mathematics

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Example: Moody Mathematics

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

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

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**So: We will just start “In the same circle. …” where the**

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. Moody Mathematics

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**In the same circle,* Congruent central angles...**

...intercept congruent arcs. Moody Mathematics

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**In the same circle,* Congruent Chords...**

...intercept congruent arcs. Moody Mathematics

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**Tangent segments from an exterior point to a circle…**

...are congruent. Moody Mathematics

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**The radius drawn to a tangent at the point of tangency…**

...is perpendicular to the tangent. Moody Mathematics

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**If a diameter (or radius) is perpendicular to a chord, then…**

...it bisects the chord… …and the arcs. Moody Mathematics

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**In the same circle,* Congruent Chords...**

...are equidistant from the center. Moody Mathematics

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**Example: Given a circle of radius 5” and two 8” chords**

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

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**If two Inscribed angles intercept the same arc...**

...then they are congruent. Moody Mathematics

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**If an inscribed angle intercepts or is inscribed in a semicircle …**

...then it is a right angle. Moody Mathematics

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**If a quadrilateral is inscribed in a circle then each pair of opposite angles …**

...must be supplementary. (total 180o) Moody Mathematics

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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: Moody Mathematics

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

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Example: Moody Mathematics

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

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Example: Moody Mathematics

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Let’s Practice!

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Example: Given Moody Mathematics

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Example: Moody Mathematics

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Example: Moody Mathematics

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Example: Moody Mathematics

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**Example: Given a circle of radius 13” and two 24” chords**

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

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Example: Moody Mathematics

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Example: Moody Mathematics

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Example: Moody Mathematics

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Example: Moody Mathematics

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Example: Moody Mathematics

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Example: Moody Mathematics

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**Example: Of the following quadrilaterals, which can not always be inscribed in a circle?**

Rectangle Rhombus Square Isosceles Trapezoid

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Example: Moody Mathematics

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Example: Moody Mathematics

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**Example: Regular Hexagon ABCDEF is inscribed in a circle.**

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THE END! Now go practice!

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