GEOMETRY
OBJECTIVES/ASSIGNMENT
Use properties of arcs of circles, as applied.
Use properties of chords of circles.
Assignment: pp. 607-608 #3-47
Reminder Quiz after 10.3 and 10.5
USING ARCS OF CIRCLES
In a plane, an angle
whose vertex is the
center of a circle is a
central angle of the
circle. If the measure
of a central angle,
APB is less than
180°, then A and B
and the points of P
central angle
A
major
arc
minor
arc
P
B
C
USING ARCS OF CIRCLES
in the interior of APB
form a minor arc of the
circle. The points A and
B and the points of
P
in the exterior of APB
form a major arc of the
circle. If the endpoints of
an arc are the endpoints
of a diameter, then the
arc is a semicircle.
central angle
A
major
arc
minor
arc
P
B
C
NAMING ARCS
G
Arcs are named by their
endpoints. For
example, the minor arc
associated with APB
above is
. MajorAB
arcs and semicircles
are named by their
endpoints and by a
point on the arc.

60°
60°
E
H
F
E
180°
NAMING ARCS
G
For example, the major
arc associated with
APB is
.
Ahere
CB on
GF is a semicircle.
theEright
The measure of a minor
arc is defined to be the
measure of its central
angle.


60°
60°
E
H
F
E
180°
NAMING ARCS

For instance, m
=GF
mGHF = 60°.
m GF
is read “the
measure of arc GF.”
You can write the
measure of an arc next
to the arc. The measure
of a semicircle is always
180°.

G
60°
60°
E
H
F
E
180°
NAMING ARCS

G
The measure of a major
GF
arc is defined as the
difference between 360° E
and the measure of its
associated minor arc.
For example, m
=GEF
360° - 60° = 300°. The
measure of the whole
circle is 360°.
60°
60°

H
F
E
180°
EX. 1: FINDING MEASURES OF ARCS
Find the measure of
each arc of R.
a. MN
b. MPN
c. PMN


N
80°
R
M
P
EX. 1: FINDING MEASURES OF ARCS
Find the measure of
each arc of R.
a. MN
b. MPN
c. PMN
Solution:
MN is a minor arc, so m
MN = mMRN =
80°


 
N
80°
R
M
P
EX. 1: FINDING MEASURES OF ARCS
Find the measure of
each arc of R.
a. MN
b. MPN
c. PMN
Solution:
MPN is a major arc, so m
MPN = 360° – 80°
= 280°


 
N
80°
R
M
P
EX. 1: FINDING MEASURES OF ARCS
Find the measure of
each arc of R.
a. MN
b. MPN
c. PMN
Solution:
PMN
is a semicircle, so
m PMN= 180°


 
N
80°
R
M
P
NOTE:
C
A
Two arcs of the same circle
are adjacent if they
intersect at exactly one
point. You can add the
measures of adjacent
areas.
Postulate 26—Arc Addition
Postulate. The measure of
an arc formed by two
adjacent arcs is the sum of
the measures of the two
arcs.

 
B
m ABC = m AB + m BC
EX. 2: FINDING MEASURES OF ARCS
Find the measure of each
arc.
a. GE
b. G EF
c. GF
m GE= m GH+ m HE =
40° + 80° = 120°
G

  
H
40°
80°
R
110°
F
E
EX. 2: FINDING MEASURES OF ARCS
Find the measure of each
arc.
a. GE
b. G EF
c. GF
m G EF= m GE+ m EF =
120° + 110° = 230°
G

  
H
40°
80°
R
110°
F
E
EX. 2: FINDING MEASURES OF ARCS
Find the measure of each
arc.
a. GE
b. G EF
c. GF
m GF = 360° - m G EF=
360° - 230° = 130°


G
H
40°

80°
R
110°
F
E
EX. 3: IDENTIFYING CONGRUENT
ARCS
A
Find the measures of
the blue arcs. Are the
arcs congruent?
•
 
 
AB and DC are in
the same circle and
m AB = m DC= 45°.
So, AB  DC
D
B
45°
45°
C
EX. 3: IDENTIFYING CONGRUENT
ARCS
Find the measures of
the blue arcs. Are the
arcs congruent?
 
 
• PQ and RS are in
congruent circles and
m PQ = m RS = 80°.
So, PQ  RS
80°
Q
P
80°
S
R
EX. 3: IDENTIFYING CONGRUENT
ARCS
Find the measures of the
blue arcs. Are the arcs
congruent?


 

Z
X
• m XY = m ZW = 65°, but
XYand ZW are not arcs of the
same circle or of
congruent circles, so XY
and ZW are NOT
congruent.

65°
Y
W
USING CHORDS OF CIRCLES
A point Y is called the
midpoint of
if
Z
 X.YAny Y
line,
segment, or ray that
contains Y bisects
.
XYZ
 

THEOREM 10.3
In the same circle, or in
congruent circles, two
minor arcs are congruent if
and only if their
corresponding chords are
congruent.

AB  BC if and only if
AB  BC
A
C
B
THEOREM 10.5
If a diameter of a circle
is perpendicular to a
chord, then the
diameter bisects the
chord and its arc.
F
E
 
G
DE  EF ,
DG  GF
D
THEOREM 10.4
If one chord is a
perpendicular bisector
of another chord, then
the first chord is a
diameter.
J
M
JK is a diameter of
the circle.
K
L
EX. 4: USING THEOREM 10.4
(x + 40)°
You can use Theorem
10.4 to find m
.

2x°
C
AD


 
A
• Because AD  DC,
and AD  DC . So,
m AD = m DC
2x = x + 40
x = 40
B
Substitute
Subtract x from each
side.
EX. 5: FINDING THE CENTER OF A
CIRCLE
Theorem 10.6 can
be used to locate a
circle’s center as
shown in the next
few slides.
Step 1: Draw any
two chords that are
not parallel to each
other.
EX. 5: FINDING THE CENTER OF A
CIRCLE
Step 2: Draw the
perpendicular
bisector of each
chord. These are
the diameters.
EX. 5: FINDING THE CENTER OF A
CIRCLE
Step 3: The
perpendicular
bisectors intersect
at the circle’s
center.
center
EX. 6: USING PROPERTIES OF
CHORDS
Masonry Hammer. A
masonry hammer has a
hammer on one end and
a curved pick on the
other. The pick works
best if you swing it along
a circular curve that
matches the shape of the
pick. Find the center of
the circular swing.
EX. 6: USING PROPERTIES OF
CHORDS
Draw a segment AB, from
the top of the masonry
hammer to the end of the
pick. Find the midpoint C,
and draw perpendicular
bisector CD. Find the
intersection of CD with
the line formed by the
handle. So, the center of
the swing lies at E.
THEOREM 10.7
In the same circle, or
in congruent circles,
two chords are
congruent if and only
if they are
equidistant from the
center.
AB  CD if and only if
EF  EG.
C
G
D
E
B
F
A
EX. 7: USING THEOREM 10.7
AB = 8; DE = 8, and CD
= 5. Find CF.
A
8 F
B
C
E
5
8
G
D
EX. 7: USING THEOREM 10.7
Because AB and DE are
congruent chords, they
are equidistant from
the center. So CF  CG.
To find CG, first find
DG.
CG  DE, so CG bisects
DE. Because DE = 8,
DG = =4.
8
2
A
8 F
B
C
E
5
8
G
D
EX. 7: USING THEOREM 10.7
Then use DG to find CG.
DG = 4 and CD = 5, so
∆CGD is a 3-4-5 right
triangle. So CG = 3.
Finally, use CG to find
CF. Because CF 
CG, CF = CG = 3
A
8 F
B
C
E
5
8
G
D
REMINDERS:
Quiz after 10.3
Last day to check Vocabulary from Chapter 10 and Postulates/Theorems from
Chapter 10.