Arcs of a Circle

Arc: Consists of two points on a circle and all points needed to connect the points by a single path. The center of an arc is the center of the circle of which the arc is a part.

Central Angle: An angle whose vertex is at the center of a circle. Radii OA and OB determine central angle AOB.

Minor Arc: An arc whose points are on or between the side of a central angle. Central angle APB determines minor arc AB. Minor arcs are named with two letters. Major Arc: An arc whose points are on or outside of a central angle. Central angle CQD determines major arc CFD. Major arcs are named with three letters (CFD).

Semicircle: An arc whose endpoints of a diameter. Arc EF is a semicircle.

Measure of an Arc Minor Arc or Semicircle: The measure is the same as the central angle that intercepts the arc. Major Arc: The measure of the arc is 360 minus the measure of the minor arc with the same endpoints.

Congruent Arcs Two arcs that have the same measure are not necessarily congruent arcs. Two arcs are congruent whenever they have the same measure and are parts of the same circle or congruent circles.

Theorems of Arcs, Chords & Angles… Theorem 79: If two central angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are congruent. Theorem 80: If two arcs of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.

Theorems of Arcs, Chords & Angles… Theorem 81: If two central angles of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent. Theorem 82: If two chords of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.

Theorems of Arcs, Chords & Angles… Theorem 83: If two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent. Theorem 84: If two chords of a circle (or of congruent circles) are congruent, then the corresponding arcs are congruent.

If the measure of arc AB = 102 º in circle O, find m  A and m  B in Δ AOB. 1.Since arc AB = 102º, then  AOB = 102º. 2.The sum of the measures of the angles of a trianlge is 180 so… 1.m  AOB + m  A + m  B = m  A + m  B = m  A + m  B = 78 3.OA = OB, so  A   B 4.m  A = 39 & m  B = 39.

1.Circles P & Q 2.  P   Q 3.RP  RQ 4.AR  RD 5.AP  DQ 6.Circle P  Circle Q 7.Arc AB  Arc CD 1.Given 2.Given Given 5.Subtraction Property 6.Circles with  radii are . 7.If two central  s of  circles are , then their intercepted arcs are .