AMSCO'S Geometry. New York - Rye High School

594 Geometry of the CirclePostulatesTheorems andCorollaries•A tangent to a circle is a line in the plane of the circle that intersects thecircle in one and only one point.•A secant of a circle is a line that intersects the circle in two points.•A common tangent is a line that is tangent to each of two circles.•A tangent segment is a segment of a tangent line, one of whose endpointsis the point of tangency.13.1 If ABX and BCX are two arcs of the same circle having a common endpointand no other points in common, then ABX BCX ABC X andmABX mBCX m ABC X .(Arc Addition Postulate)13.2 At a given point on a given circle, one and only one line can be drawnthat is tangent to the circle.13.1 All radii of the same circle are congruent.13.2 In a circle or in congruent circles, central angles are congruent if andonly if their intercepted arcs are congruent.13.3 In a circle or in congruent circles, two chords are congruent if and onlyif their central angles are congruent.13.4 In a circle or in congruent circles, two chords are congruent if and onlyif their arcs are congruent.13.5 A diameter perpendicular to a chord bisects the chord and its arcs.13.5a A line through the center of a circle that is perpendicular to a chordbisects the chord and its arcs.13.6 The perpendicular bisector of the chord of a circle contains the center ofthe circle.13.7 Two chords are equidistant from the center of a circle if and only if thechords are congruent.13.8 In a circle, if the lengths of two chords are unequal, the shorter chord isfarther from the center.13.9 The measure of an inscribed angle of a circle is equal to one-half themeasure of its intercepted arc.13.9a An angle inscribed in a semicircle is a right angle.13.9b If two inscribed angles of a circle intercept the same arc, then they arecongruent.13.10 A line is tangent to a circle if and only if it is perpendicular to a radiusat its point of intersection with the circle.13.11 Tangent segments drawn to a circle from an external point are congruent.13.11a If two tangents are drawn to a circle from an external point, then the linesegment from the center of the circle to the external point bisects theangle formed by the tangents.13.11b If two tangents are drawn to a circle from an external point, then the linesegment from the center of the circle to the external point bisects theangle whose vertex is the center of the circle and whose rays are the tworadii drawn to the points of tangency.