Download 10.2 Congruent Chords

10.3 Arcs Of A Circle
After studying this section, you will be able to
identify the different types of arcs, determine
the measure of an arc, recognize congruent
arcs, apply the relationships between
congruent arcs, chords, and central angles.
Types of Arcs
Definition
An arc consists of two points on a circle
and all points on the circle needed to
connect the points by a single path.
Definition
The center of an arc is the center of the
circle of which the arc is a part.
Definition
A central angle is an angle whose vertex is
at the center of a circle.
A
B
O
Definition
A minor arc is an arc whose points are on
or between the sides of a central angle.
Definition
A major arc is an arc whose points are on
or outside the sides of a central angle.
A
B
O
Definition
A semi circle is an arc whose endpoints
are the endpoints of a diameter.
B
O
A
E
The symbol
is used to label arcs.
The minor arc is named with 2 letters.
The major arc is named with 3 letters.
The middle letter helps indicate the
direction of a major arc. Can you name
a major and a minor arc?
F
X
The measure of an arc
Definition
The measure of a minor arc or a semicircle
is the same as the measure of the central
angle that intercepts the arc.
Definition
The measure of a major arc is 360 minus
the measure of the minor arc with the
same endpoints.
Example
Example
Given : AB  20
Given : XQY  110
A
B
Find : ACB C
P
Find : XDY
X
Y
Q
D
Congruent Arcs
Two arcs that have the same measure are not
A
necessarily congruent arcs.
D
E
B
O
In concentric circles above, arcs AB and DE have the
same measure but they are not congruent.
Why do you think they are not congruent?
Definition
Two arcs are congruent whenever they
have the same measure and are parts of
the same circle or congruent circles.
A
80
We may conclude AB  CD
B
O
C
80
D
E
Q
98
G
F
P
98
H
If circle P is congruent to circle Q, we may conclude that
EF  GH
Relating Congruent Arcs, Chords, and Central Angles
A
In the diagram, points A and B
determine one central angle,
one chord, and two arcs (one
major and one minor).
minor arc
B
central angle
major arc
Theorem
Theorem
If two central angles of a circle (or of
congruent circles) are congruent, then their
intercepted arcs are congruent.
If two arcs of a circle (or of congruent
circles) are congruent, then their
corresponding central angles are congruent.
Theorem
Theorem
If two central angles of a circle (or of
congruent circles) are congruent, then the
corresponding chords are congruent.
If two chords of a circle (or of congruent
circles) are congruent, then their
corresponding central angles are congruent.
Theorem
Theorem
If two arcs of a circle (or of congruent
circles) are congruent, then their
corresponding chords are congruent.
If two chords of a circle (or of congruent
circles) are congruent, then their
corresponding arcs are congruent.
Example 1
A
D
Given: Circle B
D is the midpoint of AC
Conclusion
BD bisects ABC
B
C
Example 2
A
102
If mAB  102 in circle O,
O
B
find mA and mB in circle O
Example 3
A. What fractional part of a circle is 36 ? Of 200 ?
B. Find the measure of an arc that is
7
12
of its circle.
Example 4
Given: Circles P and Q
P  Q; AR  RD
R
D
A
Prove AB  CD
P
B
C
Q
Summary
Summarize what you have learned
about arcs of a circle.
Homework: worksheet