B D O A C Aim: What is a circle? Homework: Workbook page 370 What are some special angles in a circle? How do we find the measure of angles formed by tangents, secants, and chords? How do we find the measure of line segments formed by tangents, secants, chords? Homework: Workbook page 370 #9 – 11 Do Now: In circle O, AOC and COB are supplementary. If mAOC = 2x, a COB = x+90, and m AOD = 3x+10. Find the following: a) X = e) DOB = i) arc AD = b) AOC = f) arc AC = j) arc DB = c) COB = g) arc BC = k) arc ADC = d) AOD = h) arc AB = l) arc BCD = B D O A C

Circles! A set of points a given distance away from a given point Parts of a Circle: Center- Circle O (x2 + y2 = r2 Radius (radii) - line segment with one endpoint at the circle’s center and the other endpoint on the circle Chord- a line segment with both endpoints on the circle Diameter- a chord that passes through the center of the circle Tangent- a line that touches the circle exactly once Secant- a line that touches the circle twice; a chord that extends outside the circle Arc- part of the circle 0 < minor arc < 180 180 < major arc < 360 *Always assume the minor arc unless states the major arc T P S R Q O

Given- Circle O PA is tangent to circle O PB is tangent to circle O Prove- PA  PB B A O P Two tangents drawn to a circle from the same external points are congruent.

-measure of the angle is equal to the measure of the intercepted arc Central Angles -formed by two radii -measure of the angle is equal to the measure of the intercepted arc -congruent arcs have congruent angles Inscribed Angles -formed by two chords meeting on the circle -the measure of the inscribed angle is half the measure of the intercepted arc -congruent arcs have congruent inscribed angles 40 50

x = ½ a x = ½ (a – b) An angle formed by a tangent and a chord: 180 An angle formed by two secants: 180 A x x = ½ a x = ½ (a – b) B X A

a + b = 360 x = ½ (a – b) x = ½ (a – b) An angle formed by two tangents: An angle formed by a secant and a tangent: a + b = 360 x = ½ (a – b) X B A x = ½ (a – b) X B A

x = ½ (a + b) An angle formed by two chords: (cross cords angle) B X

If two chords intersect in a circle, the product of the segments of one chord is equal to the product of the segments of the other. If a tangent and a secant are drawn to a circle from the same external point, the product of the secant and its external segment is equal to the tangent squared. X 9 A D 8 B C 12 a  b = c  d X X Z 3 Y 9 x2 = z (y + z)

If two secants are drawn to a circle from the same external point, the product of one secant and its external segment is equal to the product of the other secant and its external segment. 8 R M 3 P x N x - 2 r (m + r) = p (n + p)

Circle O with tangent MN 1- 31 2- 64.5 3- 54.5 4- 30 5- 34.5 6- 61 7- 30 8- 95.5 9- 84.5 10- 95.5 11- 84.5 12- 54.5

Circle O with Tangent 1- 20 2- 70 3- 70 4- 40 5- 20 6- 70 7- 90 8- 50 9- 40 10- 50 11- 20 12- 120 13- 60 14- 120 15- 60 15- 60 16- 40 17- 50 18- 90