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10.2Honors.notebook March 25, 2012 Arcs Angles and Arcs Section 10.2: Arcs and Chords 1. I can define arcs of a circle and apply their properties to solve problems. 2. I define chords of circles and apply their properties to solve problems. A B D • The sum of the measures of the central angles of a circle is 360 degrees. • A central angle separates the circle into parts, each of which is an arc. • Arc - a part or portion of a circle that is defined by two endpoints. c Textbook HW Section 10.2 Arcs of a Circle: Graphic Organizer Type of Arc: Minor Arc A Example: Major Arc Semicircle D J 1. (HW# 12) What of arc is arc PQ? a. minor arc b. major arc C 60 B M G F Named: Arc Degree Measure Equals Usually by the letters of the two endpoints. Ex: By the letters of the two endpoints and another point on the arc. Ex: N K L By the letters of the two endpoints and another point on the arc. Ex: The measure of the 360 minus the measure 360 ÷2 or 180 degrees central angle. Less of the minor arc. than 180 degrees. Greater than 180 degrees. A P C central angle B major arc - an arc with a measure of more than 180 degrees. minor arc - an arc with a measure of less than 180 degrees. The measure of each arc is related to the measure of its central angle. Postulate 26: c. semicircle 2. (HW#14) What type of arc is arc PQT? a. minor arc b. major arc c. semicircle E 110 Arcs are usually denoted by this symbol_______. AB Central angle - an angle that intersects a circle in two points and has its vertex at the center. The sides of the central angle contain two radii of the circle. 3. (HW#16) What type of arc is arc TUQ? a. minor arc b. major arc c. semicircle 4. (HW#20) What is the measure of arc KL? 5. (HW#22) What is the measure of arc LNK? Two arcs of the same circle are adjacent if they intersect at exactly one point. A C Arc Addition Postulate: The B measure of an arc formed by two adjacent arcs is the sum of the measure of the two arcs. __________________________ 6. (HW#24) What is the measure of arc NJK? 7. On a scale of 1 5, with one being the highest, how confident are you with the statement: I can identify arcs of circles and their measures? 1 10.2Honors.notebook March 25, 2012 Measures of Arcs using Arc Addition Postulate: Find the measure of each arc: C A Arc AB ____________ 40 E Arc ABD ___________ 80 110 B Arc AD _____________ Congruent Arcs Remember... how do we determine if two circles are congruent? congruent arcs - two arcs of the same circle or of congruent circles are congruent arcs if their measures are the same. • Two minor arcs, of the same circle or of congruent circles, are congruent if their central angles are congruent. 60 60 3. (X+40) C A Find the measure of arc AD. B In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. A C B If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. F E G D Z Theorem 10.6 W D Theorem 10.5 2. Example Theorem Theorem 10.4 45 C Y 2X 45 Are these arcs congruent? 65 Example: Theorem 10.4 D B X D Theorems About Chords and Circles A 1. If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. M J K L Homework Problems: Page 608: 32 - 46 even 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. C G E A F D B Example: CD = 40 ED = 25 Find EF = 2