Arcs of a Circle

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Arcs of a Circle
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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.
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Central Angle An angle whose vertex is at the
center of a circle. Radii OA and OB determine
central angle AOB.
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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).
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Semicircle An arc whose endpoints of a
diameter. Arc EF is a semicircle.
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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.

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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.

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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.

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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.

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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.

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If the measure of arc AB 102º in circle O, find
m?A and m?B in ?AOB.

1. Since arc AB 102º, then ?AOB 102º.
2. The sum of the measures of the angles of a
   trianlge is 180 so
3. m?AOB m?A m?B 180
4. 102 m?A m?B 180
5. m?A m?B 78
6. OA OB, so ?A ? ?B
7. m?A 39 m?B 39.

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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
3. .
4. Given
5. Subtraction Property
6. Circles with ? radii are ?.
7. If two central ?s of ? circles are ?, then their
   intercepted arcs are ?.