11.2.2 Angle Relationships in Circles Key Objectives • Find the measures of angles formed by lines that intersect circles. • Use angle measures to solve problems. Theorems, Postulates, Corollaries, and Properties • Theorem If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. • Theorem If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. • Theorem If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs. Example 1 Using Tangent-Secant and Tangent-Chord Angles If a tangent and a secant (or chord) intersect on a
circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. Compare this theorem to the Inscribed Angles Theorem. The measure of an angle between a tangent and a secant to a circle is determined in this example. The measure of the intercepted arc is given. The tangent and the secant intersect the circle at the point of tangency, so the measure of ∠ PQR is equal to one-half the measure of the arc it intercepts. Substituting the given value and simplifying gives m ∠ PQR = 20 ° . The measure of an arc intercepted by a tangent and a chord to a circle is determined in this example. It is given that the chord is a diameter. The tangent and the chord intersect the circle at the point of tangency. Since the chord is a diameter the angle at the intersection is a right angle. SRQ m is equal to twice the measure of the angle that intercepts it. Substituting the given value and simplifying gives = SRQ m 180 .
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