GeometryGeometry 10.2 Finding Arc Measures 2/24/2010

Geometry Geometry Objectives/Assignment Use properties of arcs of circles, as applied.

Geometry Geometry Using Arcs of Circles In a plane, an angle whose vertex is the center of a circle is a central angle of the circle. If the measure of a central angle,  APB is less than 180 °, then A and B are the points of P

Geometry Geometry Using Arcs of Circles The interior of  APB form a minor arc of the circle. The points 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. Minor Arc: < 180˚ Major Arc: > 180˚

Geometry Geometry Naming Arcs Arcs are named by their endpoints. Major arcs and semicircles are named by their endpoints and by a point on the arc. (3 Letters). The major arc is named The minor arc is named

Geometry Geometry Naming Arcs Name the semicircle. The measure of a minor arc is defined to be the measure of its central angle. 60 °

Geometry Geometry Naming Arcs For instance, m = m  GHF = 60 °. m is read “the measure of arc GF.” You can write the measure of an arc next to the arc. The measure of a semicircle is always 180°. 60 ° 180 °

Geometry Geometry Naming Arcs The measure of the whole circle is 360°. The measure of a major arc is defined as the difference between 360° and the measure of its associated minor arc. Find m. 60 ° 180 °

Geometry Geometry Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. 80 °

Geometry Geometry Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: is a minor arc, so m = m  MRN = 80 ° 80 °

Geometry Geometry Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: is a major arc, so m = 360 ° – 80 ° = 280 ° 80 °

Geometry Geometry Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: is a semicircle, so m = 180 ° 80 °

Geometry Geometry Note: Two arcs of the same circle are adjacent if they intersect at exactly one point. You can add the measures of adjacent areas. 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 = m + m

Geometry Geometry Ex. 2: Finding Measures of Arcs Find the measure of each arc. a. b. c. m = m + m = 40 ° + 80° = 120° 40 ° 80 ° 110 °

Geometry Geometry Ex. 2: Finding Measures of Arcs Find the measure of each arc. a. b. c. m = m + m = 120 ° + 110° = 230° 40 ° 80 ° 110 °

Geometry Geometry Ex. 2: Finding Measures of Arcs Find the measure of each arc. a. b. c. m = 360 ° - m = 360 ° - 230° = 130° 40 ° 80 ° 110 °