Download Answer - BakerMath.org

10.2 Angles and Arcs
Objectives


Recognize major arcs, minor arcs,
semicircles, and central angles and
their measures
Find arc length
Angles and Arcs
Sum of Central s and Arcs = 360°
Angles and Arcs




The sum of the measures of the central
angles is 360°.
A minor arc is less then 180° and is labeled
using the two endpoints.
A major arc is greater than 180° but less
than 360° and is labeled using the two
endpoints and another point on the arc.
A semicircle measures 180° and is labeled
using the two endpoints and another point
on the arc.
Angles and Arcs


Theorem 10.2: In the same circle or
 circles, two arcs are  if their
corresponding central angles are .
Postulate 10.1: The measure of an
arc formed by two adjacent arcs is the
sum of the measures of the arcs.
Example 1a:
ALGEBRA Refer to
Find
.
.
Example 1a:
The sum of the measures of
Substitution
Simplify.
Add 2 to each side.
Divide each side
by 26.
Use the value of x to find
Given
Substitution
Answer: 52
Example 1b:
ALGEBRA Refer to
Find
.
.
Example 1b:
form a linear pair.
Linear pairs are supplementary.
Substitution
Simplify.
Subtract 140 from each side.
Answer: 40
Your Turn:
ALGEBRA Refer to
a. Find m
Answer: 65
b. Find m
Answer: 40
.
Example 2a:
In
Find
bisects
.
and
Example 2a:
is a minor arc, so
is a semicircle.
is a right angle.
Arc Addition Postulate
Substitution
Subtract 90 from each side.
Answer: 90
Example 2b:
In
Find
bisects
.
and
Example 2b:
since
bisects
.
is a semicircle.
Arc Addition Postulate
Subtract 46 from each side.
Answer: 67
Example 2c:
In
Find
bisects
.
and
Example 2c:
Vertical angles are congruent.
Substitution.
Substitution.
Subtract 46 from each side.
Substitution.
Subtract 44 from each side.
Answer: 316
Your Turn:
In
and
bisects
a.
Answer: 54
b.
Answer: 72
c.
Answer: 234
are diameters,
Find each measure.
and
Example 3a:
BICYCLES This graph shows the percent of each type
of bicycle sold in the United States in 2001.
Find the measurement of the central angle representing
each category. List them from least to greatest.
Example 3a:
The sum of the percents is 100% and represents the
whole. Use the percents to determine what part of the
whole circle
each central angle contains.
Answer:
Example 3b:
BICYCLES This graph shows the percent of each type
of bicycle sold in the United States in 2001.
Is the arc for the wedge named Youth congruent to the
arc for the combined wedges named Other and
Comfort?
Example 3b:
The arc for the wedge named Youth represents 26%
or
of the circle. The combined wedges named
Other and Comfort represent
. Since
º, the arcs
are not congruent.
Answer: no
Your Turn:
SPEED LIMITS This graph shows the percent of U.S.
states that have each speed limit on their interstate
highways.
Your Turn:
a. Find the measurement of the central angles
representing each category. List them from least to
greatest.
Answer:
b. Is the arc for the wedge for 65 mph congruent to the
combined arcs for the wedges for 55 mph and 70 mph?
Answer: no
Arc Length

Another way to measure an arc is by its
length. An arc is part of a circle, so its
length is part of the circumference. We use
proportions to solve for the arc length, l.
degree measure of arc = arc length
degree measure of
circumference
Example 4:
In
and
. Find the length of
In
and
Write a proportion to compare each part to its whole.
.
.
Example 4:
degree measure of arc
degree measure of
whole circle
arc length
circumference
Now solve the proportion for .
Multiply each side by 9 .
Simplify.
Answer: The length of
is
units or about 3.14 units.
Your Turn:
In
and
. Find the length of
Answer:
units or about 49.48 units
.
Assignment


Geometry
Pg. 533 #14 – 37, 40, 47 - 50
Pre-AP Geometry
Pg. 533 #14 – 43, 47 - 52