# CIRCLES 2 Moody Mathematics.

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CIRCLES 2 Moody Mathematics

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

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

Example: Moody Mathematics

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

Example: Moody Mathematics

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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...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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…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

Example: Moody Mathematics

…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

Example: Moody Mathematics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Example: Moody Mathematics

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

Let’s Practice!

Example: Given Moody Mathematics

Example: Moody Mathematics

Example: Moody Mathematics

Example: Moody Mathematics

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

Example: Moody Mathematics

Example: Moody Mathematics

Example: Moody Mathematics

Example: Moody Mathematics

Example: Moody Mathematics

Example: Moody Mathematics

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

Example: Moody Mathematics

Example: Moody Mathematics

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

THE END! Now go practice!