10.2 Arcs and Chords Unit IIIC Day 3

Do Now List some things you know about tangent lines in relation to circles.

Using Arcs of Circles Central angle: an angle whose vertex is the center of a circle

Central Angles Let central angle APB be less than 180°. A and B and the points of P in the interior of APB form a minor arc of the circle. A and B and the points of P in the exterior of APB form a major arc of the circle. If the endpoints of an arc are the endpoints of a diameter, then the arc is a ______________. semicircle

Naming Arcs Arcs are named by their endpoints. The minor arc associated with GHF above is ____. Major arcs and semicircles are named by their endpoints and by another point on the arc (three letters). Ex.: _____

Arc Measure The measure of a minor arc is defined to be the measure of its central angle. m = mGHF = _____. m = ______ 60º read “the measure of arc GF.” The measure of a major arc is defined as the difference between 360° and the measure of its associated minor arc.

Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. m arcMN is a minor arc, so m arc MN = mMRN = 80º MPN is a major arc so m arc MPN = 360 – 80 = 280º PMN is a semicircle, so m arc PMN = 180º

Adjacent Arcs Two arcs of the same circle are adjacent if they intersect at exactly one point. Postulate 26—Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. m =

Ex. 2: Finding Measures of Arcs Find the measure of each arc. m m arc GE = m arc GH + m arc HE = 40 + 80 = 120º m arc GEF = m arc GE + m arc EF = 120 + 110 = 230º m arc GF = 360 – m arc GEF = 360 – 230 = 130º

Congruent Arcs Two arcs of the same circle or of congruent circles are congruent arcs if they have the same measure. So, two minor arcs of the same circle or of congruent circles are congruent if _____________________________. They have the same central angle measure.

Ex. 3: Identifying Congruent Arcs Find the measures of the blue arcs. Are the arcs congruent? a) AB and DC are in the same circle. m arc AB = m arc DC = 45º. So arc AB  arc DC. b) PQ and RS are in congruent circles. mPQ = mRS = 80º. So arc PQ  arc RS. c) mXY = mZW = 65º. But XY and ZW are not in the same circle or congruent circles. So XY and ZW are not congruent.