AMSCO'S Geometry. New York - Rye High School

568 Geometry of the CircleWe can state what we have just proved on page 567 as a theorem.Theorem 13.12The measure of an angle formed by a tangent and a chord that intersect atthe point of tangency is equal to one-half the measure of the intercepted arc.DBOEACAngles Formed by Two Intersecting ChordsWe can find how the measures of other angles and their intercepted arcs arerelated. For example, in the diagram, two chords AB and CD intersect in theinterior of circle O and DB is drawn.Angle AED is an exterior angle of DEB.Therefore,mAED mBDE mDBE1 X2 mBC 112 mDAX1 X2Notice that BCX(mBC 1 mDA X )is the arc intercepted by BEC and DAX is the arc interceptedby AED, the angle vertical to BEC. We can state this relationship asa theorem.Theorem 13.13The measure of an angle formed by two chords intersecting within a circle isequal to one-half the sum of the measures of the arcs intercepted by theangle and its vertical angle.Angles Formed by Tangents and SecantsWe have shown how the measures of angles whose vertices are on the circle orwithin the circle are related to the measures of their intercepted arcs. Now wewant to show how angles formed by two tangents, a tangent and a secant, or twosecants, all of which have vertices outside the circle, are related to the measuresof the intercepted arcs.A Tangent Intersecting a SecantgIn the diagram, PRS is a tangent to circle O at Rgand PTQ is a secant that intersects the circle at Tand at Q. Chord RQ is drawn. Then SRQ is anexterior angle of PRQ.QSROTP