1. Find x and y. 2. ANSWER x = 60; y = 60 ANSWER x = 35; y = 35

Apply angle relationships in circles. Target Apply angle relationships in circles. You will… Use angle measures to find arc measures.

Vocabulary central angle (of a circle) – an angle whose vertex is the center of the circle and whose sides intersect the circle in exactly two points minor arc central angle minor arc – intercepted by a central angle less than 180º and in its interior arc measure = central angle measure major arc – intercepted by a central angle less than 180º and in its exterior arc measure = 360 – related minor arc semicircle – arc whose endpoints are endpoints of a diameter semicircle measure = 180º major arc

Vocabulary adjacent arcs – two arcs of the same circle with a common endpoint Arc Addition Postulate 23 – the measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs

EXAMPLE 1 Find measures of arcs Find the measure of each arc of P, where RT is a diameter. RS a. RTS b. RST c. SOLUTION 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.

EXAMPLE 2 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. Survey a. mAC SOLUTION a. mAC mAB = + mBC = 29o + 108o = 137o

EXAMPLE 2 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. Survey b. mACD SOLUTION b. mACD = mAC + mCD = 137o + 83o = 220o

EXAMPLE 2 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. Survey c. mADC SOLUTION mADC mAC = 360o – c. = 360o – 137o = 223o

EXAMPLE 2 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. Survey d. mEBD SOLUTION d. mEBD = 360o – mED = 360o – 61o = 299o

GUIDED PRACTICE for Examples 1 and 2 Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc. 1 . TQ . QRT 2 . TQR 3 . QS 4 . TS 5 . RST 6 SOLUTIONS minor arc measures: m TQ = 120o, m QS = 160o , m TS = 80o. major arc measures: m QRT= 240o . semicircles: m TQR = 180o, m RST = 180o.

EXAMPLE 3 Identify congruent arcs Tell whether the red arcs are congruent. Explain why or why not. a. b. SOLUTION 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.

EXAMPLE 3 Identify congruent arcs Tell whether the red arcs are congruent. Explain why or why not. c. SOLUTION c. VX YZ because they are in congruent circles and mVX = mYZ .

EXAMPLE 3 Identify congruent arcs Tell whether the red arcs are congruent. Explain why or why not. 8. 7. SOLUTIONS AB CD because they are in congruent circles and mAB = mCD . 7. MN and PQ have the same measure, but are not congruent because they are arcs of circles that are not congruent. 8.