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**Classifying Angles with Circles**

Case 1: Vertex is on the circle. a b.

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**Classifying Angles with Circles**

Case 2: Vertex is inside the circle.

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**Classifying Angles with Circles**

Case 3: Vertex is outside the circle. a b.

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c.

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Theorem If a tangent and a secant (or a chord) intersect ON a circle at the point of tangency, then the measure of the angle formed is one half the measure of its intercepted arc.

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**The angle on the circle is half the measure of the intercepted arc.**

75°

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Theorem The measure of an angle formed by two secants or chords that intersect in the interior of a circle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

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**It’s the average of the two intercepted arcs –**

(80+40)/2 = 60°

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Theorem The measure of an angle formed by two secants intersecting in the exterior of a circle is one-half the difference of the measures of the intercepted arcs.

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**It’s half the difference: (80-20)/2 = 30°**

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