11 2 arcs and central angles lesson
- 1. T S Arcs and Central AnglesArcs and Central Angles A ____________ is formed when the two sides of an angle meet at the center of a circle. central angle R central angle Each side intersects a point on the circle, dividing it into ____ that are curved lines. arcs There are three types of arcs: A _________ is part of the circle in the interior of the central angle with measure less than 180°. minor arc A _________ is part of the circle in the exterior of the central angle. major arc __________ are congruent arcs whose endpoints lie on the diameter of the circle. Semicircles
- 2. K P G R Arcs and Central AnglesArcs and Central Angles Types of Arcs semicircle PRTmajor arc PRGminor arc PG K P G m PG< 180 m PRG> 180 m PRT = 180 Note that for circle K, two letters are used to name the minor arc, but three letters are used to name the major arc and semicircle. These letters for naming arcs help us trace the set of points in the arc. In this way, there is no confusion about which arc is being considered. K T P R G
- 3. Arcs and Central AnglesArcs and Central Angles Depending on the central angle, each type of arc is measured in the following way. Definition of Arc Measure 1) The degree measure of a minor arc is the degree measure of its central angle. 2) The degree measure of a major arc is 360 minus the degree measure of its central angle. 3) The degree measure of a semicircle is 180.
- 4. Arcs and Central AnglesArcs and Central Angles In P, find the following measures: P H AM T 46° 80° m MA = APM m MA = 46° =m ATAPT m THM = 360° – (MPA + APT) APT = 80° m THM = 360° – (46° + 80°) m THM = 360° – (126°) m THM = 234°
- 5. Arcs and Central AnglesArcs and Central Angles P H AM T 46° 80° In P, AM and AT are examples of ________ arcs.adjacent Adjacent arcs have exactly one point in common. For AM and AT, the common point is __.A Adjacent arcs can also be added. Postulate 11-1 Arc Addition Postulate The sum of the measures of two adjacent arcs is the measure of the arc formed by the adjacent arcs. C QP R If Q is a point of PR, then mPQ + mQR = mPQR
- 6. Arcs and Central AnglesArcs and Central Angles R P S Q T 75° 65° In P, RT is a diameter. Find mQT. mQT + mQR = mTQR mQT + 75° = 180° mQT = 105° Find mSTQ. mSTQ + mQR + mRS = 360° mSTQ + 75° + 65° = 360° mSTQ + 140° = 360° mSTQ = 220°
- 7. Arcs and Central AnglesArcs and Central Angles B Suppose there are two concentric circles S60° C A D with ASD forming two minor arcs, BC and AD. Are the two arcs congruent? The arcs are in circles with different radii, so they have different lengths. However, in a circle, or in congruent circles, two arcs are congruent if they have the same measure. Although BC and AD each measure 60°, they are not congruent.
- 8. Arcs and Central AnglesArcs and Central Angles Theorem 11-3 In a circle or in congruent circles, two minor arcs are congruent if and only if (iff) their corresponding central angles are congruent. 60° 60° Z Y X W Q WX YZ iff mWQX = mYQZ
- 9. Arcs and Central AnglesArcs and Central Angles T W S K M R In M, WS and RT are diameters, mWMT = 125, mRK = 14. Find mRS. WMT RMS Vertical angles are congruent WMT = RMS Definition of congruent angles mWT = mRS Theorem 11-3 125 = mRS Substitution Find mKS. KS + RK = RS KS + 14 = 125 KS = 111 Find mTS. TS + RS = 180 TS + 125 = 180 TS = 55
- 10. Arcs and Central AnglesArcs and Central Angles 30 or More 10% 1 to 5 20% 6 to 10 23% 11 to 20 33% 21 to 30 14% Source: ICRs TeenEXCEL survey for Merrill Lynch Teens at Work Twenty-two percent of all teens ages 12 through 17 work either full or part-time. The circle graph shows the number of hours they work per week. Find the measure of each central angle. 1 – 5: = 72 6 – 10: = 83 11 – 20: = 119 21 – 30: = 50 30 or More: = 36 20% of 360 23% of 360 33% of 360 14% of 360 10% of 360