Section 6.1 Circles and Related Segments and Angles

1
Section 6.1Circles and Related Segments and
Angles

• A circle is the set of all points in a plane that
  are at a fixed distance from a given point known
  as the center of the circle.
• A circle is named by its center point.
• The symbol for circle is ?
• A radius is a segment that joins the center of
  the circle to a point on the circle.
• All radii of a circle are congruent
• A line segment that joins two points of a circle
  is a chord.
• A diameter of a circle is a chord that contains
  the center of the circle

2
Segments of a Circle

• What are the radii of ?Q?
• QS, QW, QV, QT
• What is the diameter?
• WT
• What is the chord?
• SW, TW
• Congruent circles are two or more circles that
  have congruent radii.

3
Relationships Concerning Circles

• An arc is a segment of a circle determined by two
  points on the circle and all the points in
  between. ABC is an arc denoted by ABC.
• A semicircle is the arc determined by a diameter
• A minor arc is part of a semicircle.
• A major arc is more than a semicircle but less
  than a circle

• Concentric Circles are coplanar circles that have
  a common center.
• Theorem 6.1.1 A radius that is perpendicular to
  a chord, bisects the chord.

O
D
A
P
C
B
4
Angles and Arcs

• A central angle of a circle is an angle whose
  vertex is the center of the circle and whose
  sides are radii.
• An intercepted arc is determined by the two
  points of intersection of the angle with the
  circle and all points of the arc in the interior
  of the angle.
• ?ROT is a central angle of ?O.
• RT is the intercepted arc.

5
Angle and Arc Relationships in the Circle

• Postulate 16 (Central Angle Postulate) In a
  circle, the degree measure of a central angle is
  equal to the degree measure of the intercepted
  arc.
• Congruent arcs are arcs with equal measure in
  either a circle or congruent circles.
• m?AOB m?COD
• m AEB mCFD

C
A
F
O
E
B
D
6

• Postulate 17 (Arc-Addition Postulate) If B
  lies between A and C on a circle, then mAB mBC
  mABC An arc is equal to the sum of its parts.
• Inscribed angle of a circle whose vertex is a
  point on the circle and whose sides are chords of
  the circle.
• ?RST is the inscribed angle with chords SR and
  ST as the sides.
• Theorem 6.1.2 The measure of an inscribed angle
  of a circle is one-half the measure of its
  intercepted arc.
• Case 1 One chord is a diameter (below)
• Case 2 The diameter lies inside the inscribed
  angle p. 281 a
• Case 3 The diameter lies outside the
  inscribed angle p. 281 b
• Proof Case 1 Draw the radius RO. ?ROS is
  isoceles
• with m?R m ?S.
• Since ?ROT is the exterior angle of ?ROS then
• m?ROT m?R m ?S 2 m?S
• Therefore, m?S ½ m?ROT.
• Since the measure of the arc is equal to the
  measure
• of the angle then by substitution m?S ½ m
  RT.

7
More on Chords and Arcs

• In a circle (or in congruent circles),
• Theorem 6.1.3 congruent minor arcs have
  congruent central angles p. 282.
• Theorem 6.1.4 congruent central angles have
  congruent arcs.
• Theorem 6.15 congruent chords have congruent
  minor (major) arcs.
• Theorem 6.16 congruent arcs have congruent
  chords.

C
A
O
P
B
D
8

• Theorem 6.1.7 Chords that are at the same
  distance from the center of a circle are
  congruent. Figure 6.18
• Theorem 6.1.8 Congruent chords are located at
  the same distance from the center of the circle.
  Figure 6.18
• Theorem 6.1.9 An angle inscribed in a
  semicircle is a right angle. Figure 6.19 p. 282
• Theorem 6.1.10 If two inscribed angles
  intercept the same arc, then these angles are
  congruent. Figure 6.20