Copyright © 2014 Pearson Education, Inc. 6 CHAPTER 6.1 Circles and Arcs Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Circles In a plane, a circle is the set of all points equidistant from a given point called the center. We name a circle by its center. Circle P is shown below. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Circles A diameter is a segment that contains the center of a circle and has both endpoints on the circle. A radius is a segment that has one endpoint at the center and the other endpoint on the circle. Congruent circles have congruent radii. A central angle is an angle whose vertex is the center of the circle. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Theorem 6.1 Therep  Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Congruent Circles If two circles are congruent, then their radii and diameters are congruent. Conversely, if the radii or diameters are congruent, then two circles are congruent. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Circles An arc is a part of a circle. One type of arc, a semicircle, is half of a circle. A minor arc is smaller than a semicircle. A major arc is larger than a semicircle. We name a minor arc by its endpoints and a major arc or a semicircle by its endpoints and another point on the arc. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Central Angle   Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Arc Measure The measure of an Arc is the number of degrees in the central angle that intercepts the arc. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Congruent Arcs Congruent arcs are arcs that have the same measure and are in the same circle or in congruent circles. Copyright © 2014 Pearson Education, Inc.

Postulate 6.2 Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Arc Measure • minor arc—The measure of a minor arc is equal to the measure of its corresponding central angle. • major arc—The measure of a major arc is the measure of the related minor arc subtracted from 360°. • semicircle—The measure of a semicircle is 180°. Copyright © 2014 Pearson Education, Inc.

Finding the Measures of Arcs What is the measure of each arc in circle O? a. b. c. d. Solution a. b. Copyright © 2014 Pearson Education, Inc.

Finding the Measures of Arcs What is the measure of each arc in circle O? a. b. c. d. Solution c. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Arc • A semicircle is half of a circle. • A minor arc is smaller than a semicircle. • A major arc is larger than a semicircle. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Inscribed Angle An angle whose vertex is on a circle and whose sides intersect the circle in two other points is called an inscribed angle. Thm 6.2 – The measure of an inscribed angle is one-half the measure of its intercepted arc. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Corollaries Corollary 6.3 – Inscribed angles that intercept the same or congruent arcs are congruent Corollary 6.4 – Every angle inscribed in a semicircle is a right angle Copyright © 2014 Pearson Education, Inc.