Chapter 7 Lesson 6 Objective: To find the measures of central angles and arcs.

Central Angles and Arcs In a plane, a circle is the set of all points. In a plane, a circle is the set of all points. The set of all points equidistant from a given point is the center. The set of all points equidistant from a given point is the center. A radius is a segment that has one endpoint at the center and the other endpoint on the circle. A radius is a segment that has one endpoint at the center and the other endpoint on the circle. A diameter is a segment that contains the center of a circle and has both endpoints on the circle. A diameter is a segment that contains the center of a circle and has both endpoints on the circle.

Congruent Circles have congruent radii. 5 m Central Angle is an angle whose vertex is the center of the circle. A B CD

Example 1 Finding Central Angles **Remember a circle measures 360°.** Sleep: Sleep: 31% of =111.6 Food: Food: 9% of =32.4 Work: Work: 20% of =72 Must Do: Must Do: 7% of =25.2 Entertainment: Entertainment: 18% of =64.8 Other: Other: 15% of =54

arc An arc is a part of a circle. Types of arcs Semicircle is half of a circle. A DAE A minor arc is smaller than a semicircle.minor arc A major arc is greater than a semicircle.major arc AB Minor arc D ADB Major arc

Example 2: Identifying Arcs Identify the following in O. 1.the minor arcs 2.the semicircles 3. the major arcs that contain point A O A C D E

Example 3: Identifying Arcs Identify the minor arcs, major arcs and semicircles in O with point A as an endpoint. O A B D E minor arcs AD, AE major arcs ADE, AED semicircles ADB, AEB

Adjacent arcs Adjacent arcsAdjacent arcs Adjacent arcs are arcs of the same circle that have exactly one point in common. Postulate 7-1 Postulate 7-1: Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. A B C mABC = mAB + mBC

Example 4: Finding the Measures of Arcs Find the measure of each arc. 58° 32° A B C D O BC BD ABC ABC is a semicircle. AB

Example 5: Finding the Measures of Arcs 56° 40° M C W X D Y Find mXY and mDXM in C. mXY = mXD + mDY mXY = =96 mDXM = mDX mDXM = mDXM = 220

Assignment pg #1-26; 40-54