11 2 arcs and central angles lesson

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  1. TSArcs and Central AnglesArcs and Central AnglesA ____________ is formed when the two sides of an angle meet at the centerof a circle.central…
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  • 1. TSArcs and Central AnglesArcs and Central AnglesA ____________ is formed when the two sides of an angle meet at the centerof a circle.central angleRcentralangleEach side intersects a point on the circle, dividing it into ____ that arecurved lines.arcsThere are three types of arcs:A _________ is part of the circle in the interior ofthe central angle with measure less than 180°.minor arcA _________ is part of the circle in the exterior ofthe central angle.major arc__________ are congruent arcs whose endpointslie on the diameter of the circle.Semicircles
  • 2. KPGRArcs and Central AnglesArcs and Central AnglesTypesofArcssemicircle PRTmajor arc PRGminor arc PGKPGm PG< 180 m PRG> 180 m PRT = 180Note that for circle K, two letters are used to name the minor arc, but three letters areused to name the major arc and semicircle. These letters for naming arcs help ustrace the set of points in the arc. In this way, there is no confusion about which arcis being considered.KTPRG
  • 3. Arcs and Central AnglesArcs and Central AnglesDepending on the central angle, each type of arc is measured in thefollowing way.Definitionof ArcMeasure1) The degree measure of a minor arc is the degree measureof its central angle.2) The degree measure of a major arc is 360 minus the degreemeasure of its central angle.3) The degree measure of a semicircle is 180.
  • 4. Arcs and Central AnglesArcs and Central AnglesIn P, find the following measures:PHAMT46°80°m MA = APMm MA = 46°=m ATAPTm 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 AnglesPHAMT46°80°In P, AM and AT are examples of ________ arcs.adjacentAdjacent arcs have exactly one point in common.For AM and AT, the common point is __.AAdjacent arcs can also be added.Postulate11-1ArcAdditionPostulateThe sum of the measures of two adjacent arcs is the measureof the arc formed by the adjacent arcs.CQPRIf Q is a point of PR, thenmPQ + mQR = mPQR
  • 6. Arcs and Central AnglesArcs and Central AnglesRPSQT75°65°In P, RT is a diameter.Find mQT.mQT + mQR = mTQRmQT + 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 AnglesBSuppose there are two concentric circlesS60°CADwith 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 theyhave the same measure.Although BC and AD each measure 60°, they are not congruent.
  • 8. Arcs and Central AnglesArcs and Central AnglesTheorem11-3In a circle or in congruent circles, two minor arcs are congruentif and only if (iff) their corresponding central angles arecongruent.60° 60°ZYXWQWX  YZiffmWQX = mYQZ
  • 9. Arcs and Central AnglesArcs and Central AnglesTWSKMRIn M, WS and RT are diameters, mWMT = 125, mRK = 14.Find mRS.WMT  RMS Vertical angles are congruentWMT = RMS Definition of congruent anglesmWT = mRS Theorem 11-3125 = mRS SubstitutionFind mKS.KS + RK = RSKS + 14 = 125KS = 111Find mTS.TS + RS = 180TS + 125 = 180TS = 55
  • 10. Arcs and Central AnglesArcs and Central Angles30 or More10%1 to 520%6 to 1023%11 to 2033%21 to 3014%Source: ICRs TeenEXCEL survey for Merrill LynchTeens at WorkTwenty-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: = 726 – 10: = 8311 – 20: = 11921 – 30: = 5030 or More: = 3620% of 36023% of 36033% of 36014% of 36010% of 360
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