10.1 Tangents to Circles Circle: the set of all points in a plane that are equidistant from a given point. Center: the point from which all points of a circle are set around. Radius: the distance from the center to a point on the circle. *two circles are congruent if they have the same radius.

10.1 Tangents to Circles Diameter: the distance across the circle passing through the center. *all radii of a circle are congruent. Chord: a segment whose end points are on the circle. (Diameter is a specific chord) Secant: a line that intersects a circle in two points. (A chord that keeps going past the sides of the circle) Tangent: a line in the plane of a circle that intersects the circle in only one point.

10.1 Tangents to Circles

10.1 Tangents to Circles Tangent circles: Circles that intersect in one point. Concentric: circles that share a common center. Common Tangent: a tangent line that is shared by two coplaner circles.

10.1 Tangents to Circles Interior of a circle: consists of the points that are inside the circle. Exterior of a circle: consists of the points that are outside the circle. Point of tangency: the point in which the tangent line intersects with the circle.

10.1 Tangents to Circles Theorem 10.1 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. If l l is tangent to Q at P, then l QP. Theorem 10.2 In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. If l QP, then l is tangent to Q.

10.1 Tangents to Circles Theorem 10.3 IF two segments from the same exterior point are tangent to the circle, then they are congruent. If SR and ST are tangent to P, then SR ST.

10.1 Tangents to Circles Homework: Page 599 10-16 even, 18-25 all, 26-30 even, 36-38 even, 46-48 even

10.2 Arcs and Chords Central Angle: an angle whose vertex is the center of a circle. Minor Arc: an arc whose measure is less than 180 ̊. Major Arc: an arc whose measure is greater than 180 ̊. Measure of an arc: the measure of the central angle of an arc. Congruent Arcs: two arcs of the same circle or of congruent circles whose measures are congruent.

10.2 Arcs and Chords Postulate 20 Arc Addition Postulate: The sum of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. mABC = mAB + mBC

10.2 Arcs and Chords Theorem 10.4 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. AB = BC if and only if AB = BC.

10.2 Arcs and Chords Theorem 10.5 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. DE = EF if and only if DG = GF.

10.2 Arcs and Chords Theorem 10.6 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. JK is a diameter of the circle.

10.2 Arcs and Chords Theorem 10.7 In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. AB = CD if and only if EF = EG.

10.2 Arcs and Chords Homework: Page 607 12-46 even

10.3 Inscribed Angles Inscribed angles: an angle whose vertex is on a circle and whose sides contain chords of the circle. Intercepted arc: the arc that lies in the interior of an inscribed angle and has endpoints on the angle. Inscribed: all of the vertices of a polygon lie on the circle Circumscribed: the circle is around the polygon.

10.3 Inscribed Angles Theorem 10.8 Measure of an inscribed angle If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. M ADB = ½ mAB

10.3 Inscribed Angles Theorem 10.9 If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

10.3 Inscribed Angles Theorem 10.10 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. B is a right angle if and only if AC is a diameter of the circle.

10.3 Inscribed Angles Theorem 10.11 A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. D, E, F, and G lie on some circle, C, if and only if m D + m F = 180 ̊.

10.3 Inscribed Angles Homework: Page 617 8-28 even

10.4 Other Angle Relationships in Circles Theorem 10.12 If a Tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intersected arc. M 1 = ½ mAB M 2 = ½ mBCA

10.4 Other Angle Relationships in Circles Theorem 10.13 If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its verticle angle M 1 = ½ (mCD + mAB), M 2 = ½ (mBC + mAD)

10.4 Other Angle Relationships in Circles Theorem 10.14 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 one half the difference of the measures of the intercepted arcs. m 1 = ½ (mBC - mAC) m 2 = ½ (mPQR - mPR) m 3 = ½(mXY – mWZ)

10.4 Other Angle Relationships in Circles Homework: Page 624 8-34 even

10.5 Segment Lengths in Circles Theorem 10.15 If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

10.5 Segment Lengths in Circles Tangent segment: a segment that is tangent to the circle at the segments endpoint. Secant Segment: a segment that passes through the circle, but still ends on the side of the circle. External segment: the portion of the secant segment that falls outside the circle.

10.5 Segment Lengths in Circles Theorem 10.16 If two secant segments share the same endpoint outside the circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of the external segment.

10.5 Segment Lengths in Circles Theorem 10.17 If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.

10.5 Segment Lengths in Circles Homework: Page 632 10-26 even