10.2 Find Arc Measures
Goal • Use angle measures to find arc measures.
Your Notes
VOCABULARY
Central angle
A central angle of a circle is an angle whose vertex is the center of the circle.
Minor arc
Part of a circle measuring less than 180º.
Major arc
Part of a circle measuring between 180° and 360°.
Semicircle
A semicircle is an arc with endpoints that are the endpoints of a diameter.
Measure of a minor arc
The measure of a minor arc is the measure of its central angle.
Measure of a major arc
The measure of a major arc is the difference between 360° and the measure of the related
minor arc.
Congruent circles
Two circles are congruent circles if they have the same radius.
Congruent arcs
Two arcs are congruent arcs if they have the same measure and they are arcs of the same
circle or of congruent circles.
MEASURING ARCS
The measure of a minor arc is the measure of its central angle. The expression mAB is
read as “the measure of arc AB.”
The measure of the entire circle is _360°_. The measure of a major arc is the difference
between _360°_ and the measure of the related minor arc. The measure of a semicircle is
_180°_.
m AB = 50°
mADB = 310°
Your Notes
Example 1
Find measures of arcs
Find the measure of each arc of C, where DF is a diameter.
a. DE
b. DFE
c. DEF

a. DE is a _minor_ arc, so m DE = m_DCE_ = _117°_.
b. DFE is a _major_ arc, so
m DFE = _360°_ – _117°_ = _243°_.
c. 
DF is a diameter, so DEF is a _semicircle_, and
mDEF = _180°_.
Checkpoint Complete the following exercise.

1. Find mRTS in the diagram at the right.
m RTS = 315°
POSTULATE 23: ARC ADDITION POSTULATE
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two
arcs.


m ABC = m _AB_
+ m_BC_
Example 2
Find measures of arcs
Money You join a new bank and divide your money several ways as shown in the circle
graph at the right. Find the indicated arc measures.
a. m BD
b. m BCD
Solution
a. m BD = m BA + m AD
= _55º_ + _105º_
= _160º_
The measure of a
minor arc is less
than 180º. The
measure of a major
arc is greater than
180º.
b. m BCD = 360° – m BD
= 360° – _160°_
= _200°_
Your Notes
Example 3
Identify congruent arcs
Tell whether the given arcs are congruent. Explain why or why not.
a. BC and DE

b. AB and CD
FG and HJ
c. 
Solution
a. BC _ DE because they are in _the same circle_ and mBC = m DE .
b. AB and CD have the same _measure_, but they are _not congruent_ because
they are arcs of circles that are _not congruent_.
c. FG __ HJ because they are in _congruent circles_ and mFG _=_+ m HJ .
Checkpoint Complete the following exercises.
2. In Example 2, find (a) m BCD and (b) m ABC.
(a) m BCD = 305°,
(b) m ABC = 195°
3. In the diagram at the right, isPQ  SR ? Explain why or why not.
PQ  SR because they are in the same circle and m PQ  m SR.
Homework
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