Download Geometry 10-2 Measuring Angles and Arcs

Geometry 10-2 Measuring Angles and Arcs
A central angle
of a circle is an angle whose
vertex is the center of the circle. The two sides of the angle
contain radii of the circle.
A nice property of central angles is the sum of the measures
of the (nonoverlapping) central angles of a circle is 360.
Find the value of x.
A
40
B
C
A minor arc
is the shorter arc
defined by a central angle.
A major arc
is the longer arc
defined by a central angle.
Just like angles, arcs are measured in degrees.
The measure of an arc is equal to the measure of its central
angle.
x = 145
A minor arc will always have a measure less than 180. A
major arc will always have a measure greater than 180. If
the measure of an arc is exactly 180, then the arc is a
semicircle.
We name a minor arc by its endpoints. We name a major
arc (or a semicircle) by its endpoints along with one other
point on the arc. We also include the arc symbol:
(
major arc
215 + x = 360
Angle C is a central angle.
An arc
is a portion of a circle defined by two endpoints.
A central angle splits the circle into two arcs.
minor arc
40 + 85 + x + 90 = 360
85
x
A
D
C
B
minor arc: AB or BA
major arc: ADB or BDA
Congruent arcs
are arcs in the same circle
(or in congruent circles) that have the same measure.
In a circle graph that uses percentages, the measure of an
arc is found by multiplying the percentage by 360.
Theorem 10.1: In the same circle (or in congruent circles),
two minor arcs are congruent if and only if their central
angles are congruent.
For example, if a section of a circle graph represents 22%,
then the arc's measure is 0.22(360) = 79.2 degrees.
G
F
1
2
J
H
If 1~
= 2, then FG ~
= HJ.
If FG ~
= HJ, then
1~
= 2.
Adjacent arcs
are arcs that have exactly one point in
common - that is, they share an endpoint and nothing else.
AB and BC are adjacent arcs.
A
B
C
Postulate 10.1 - Arc Addition Postulate: The measure on
an arc formed by two adjacent arcs is the sum of the
measures of the two arcs.
C
B
F
mAC = mAB + mBC
A
C
B
A
Arc length
is the distance between the endpoints
along an arc. We can find the arc length using ratios.
Arc Length
Angle
=
Circumference
360
A
B
3 80
L
x
=
2 ~ r 360
Find the length of AB.
L
80
=
2 ~ (3) 360
360L = 1507.964
L = 4.189
D
63
Given AC and BE are diameters,
find each measure.
mCE
180 - 63 = 117
mABD
360 - 63 - 90 = 207
E