10.2 Find Arc Measures Hubarth Geometry

. . The measures of a minor arc and a major arc depend on the central angle of the minor arc. Minor arc is less than 180. The measure of a minor arc is the measure of its central angle. The measure of a major arc is the difference of 360 and the measure of the related minor arc. A AB = 60 B . 60 . D C ADB = 360 – 60 = 300 A semicircle is an arc whose central angle measure 180. A semicircle is named by three points. Its measure is 180

. . . . Ex 1 Name and Find Measures of Arcs Name the arc and identify the type of arc. Find DF in figure a and LMN in figure b. . a. L b. G . . E K 110 . 40 M D N F a. DF is a minor arc. Its measure is 40 b. LMN is a major arc. Its measure is 360 -110 = 250

Ex 2 Find Measures of Arcs Find the measure of each arc of P, where RT is a diameter. a. RS b. RTS c. RST RS is a minor arc, so mRS = m RPS = 110o. a. RTS is a major arc, so mRTS = 360o 110o = 250o. b. – c. RT is a diameter, so RST is a semicircle, and mRST = 180o.

. . . Arc Addition Postulate Words The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Symbols mACB = mAC + mCB . A . C . B

Ex 3 Find Measures of Arcs A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures. a. mAC b. mACD c. mADC d. mEBD a. mAC mAB = + mBC b. mACD = mAC + mCD = 29o + 108o = 137o + 83o = 220o = 137o c. mADC mAC = 360o – d. mEBD = 360o – mED = 360o – 137o = 360o – 61o = 223o = 299o

Two circles are congruent circles if they have the same radius Two circles are congruent circles if they have the same radius. Two arcs are congruent arcs If they have the same measure and they are arcs of the same circle or of congruent circles. Ex 4 Identify Congruent Arcs Tell whether the red arcs are congruent. Explain why or why not. a. b. a. CD EF because they are in the same circle and mCD = mEF b. RS and TU have the same measure, but are not congruent because they are arcs of circles that are not congruent.

Practice Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc. 1. TQ 2. QRT 3. TQR 4. QS 120 240 180 160 Tell whether the red arcs are congruent. Explain why or why not. 5. 6. AB CD because they are in congruent circles and mAB = mCD . MN and PQ have the same measure, but are not congruent because they are arcs of circles that are not congruent.