Arcs, Central and Inscribed Angles

Arcs, Central and Inscribed Angles
ACTIVITY 4.2
continued
Coming Full Circle
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Close Reading, Interactive Word Wall,
Vocabulary Organizer
___
My Notes
___
4. Below is a diagram of circle O. OA and OB are called ______.
Q
A
O
ACADEMIC VOCABULARY
central angle
B
Points A and B divide the circle into two arcs. The smaller arc is known
as the minor arc AB , and the larger arc is known as major arc AQB . The
angle formed by the two radii, ∠AOB, is called a central angle of this
circle.
In general, a central angle is an angle whose vertex is at the center
of a circle and whose sides contain radii of the circle. An arc intercepted
by a central angle is the minor arc that lies in the interior of the angle.
Notice that the major arc associated with points A and B lies outside
∠AOB, while the minor arc lies in the interior of ∠AOB. AB is said to be
intercepted by ∠AOB.
By definition, the measure of a minor arc is equal to the measure
of the central angle that intercepts the minor arc. The notation for “the
measure of AB ” is m
AB .
The notation for a minor arc
requires the endpoints of the
arc, AB . The notation for a major
arc requires a point on the arc
included between the endpoints
of the arc, AQB . Semicircles are
named as major arcs.
• The measure of a minor arc
must be between 0° and
180°.
• The measure of a major arc
must be at least 180° and
less than 360°.
© 2010 College Board. All rights reserved.
The measure of a semicircle
is 180°.
Unit 4 • Circles and Constructions
287
ACTIVITY 4.2
Arcs, Central and Inscribed Angles
continued
Coming Full Circle
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Quickwrite
My Notes
TRY THESE A
___
__
Given the circle below with center C, diameters JR and KQ and
m∠RCQ = 50°. Use the definitions for central angle and intercepted arc
along with triangle properties to find each of the following.
= _____°
a. mRQ
= ______°
b. mJQ
Q
C
J
R
c. m∠CRQ = _______°
K
= ______°
d. mJQR
= ______°
e. mJKQ
_______. Write a definition for congruent arcs.
f. JK
TRY THESE B
__ ___
___
Given
the circle
___below with center C and diameters JR, KQ and PL.
___
__
PL ⊥ JR and KQ bisects ∠PCR.
P
C
J
b. m∠1 + m∠2 = ______°
= m_____°. Explain.
+ mPQ
c. mJP
= ______°
d. mJQ
e. m∠JRQ = ______°
= ______°
f. mJL
g. m∠JRL = ______°. Explain.
288
SpringBoard® Mathematics with MeaningTM Geometry
K
L
2
Q
1
R
© 2010 College Board. All rights reserved.
QR
.
a. Explain why PQ