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.