November 19, 2009 A central angle of a circle is an angle with its vertex at the center of the circle. The figurebelow illustrates central angle AOB in circle O. An arc is an unbroken part of the circle. The two points A and B on circle O above are the endpoints of two arcs. A and B and the points of circle O in the interior of angle AOB form a minor arc. Note that a minor arc is named by its endpoints: AB and is read "arc AB." A and B and the points of circle O not in the interior of AOB form a major arc. The major arc with endpoints A and B is illustrated in red below. Note that we use three letters to name a major arc: ACB, which is read"arc ACB." The measure of a minor arc is defined to be the measure of its central angle. We make a distinction here between the term 'measure' and the term 'length'. We will deal with the length of an arc in another section. In the diagram below, mAB denotes the measure of minor arc AB. In the next diagram we see that the measure of a major arc is 360 minus the measure of its associated minor arc. If A and B are the endpoints of a diameter, then the two arcs formed are called semicircles. Note that we name semicircles the same way we do major arcs; with three letters. The semicircle illustrated in red above (the one "on top.") is denoted ACB so that we know the arc has endpoints A and B and passes through point C. Adjacent arcs of a circle are arcs that have exactly one point in common. Arc AB and arc BC are adjacent arcs since they share only point B. Arc Addition Postulate: The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. Congruent arcs are arcs, in the same circle or in congruent circles, that have equal measures. In the diagram below, chord AB cuts off two arcs, arc AB and arc ATB. We call AB the minor arc, the arc of chord AB. Theorem: In the same circle or congruent circles: (1) Congruent arcs have congruent chords (2) Congruent chords have congruent arcs. Theorem: A diameter that is perpendicular to a chord bisects the chord and its arc. Theorem: In the same circle or in congruent circles: (1) Chords equally distant from the center (or centers) are congruent. (2) Congruent chords are equally distant from the center (or centers). Arc length is equal to the measure of the arc over 360, times the circumference.