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Central Angles and Inscribed Angles
Essential question: What is the relationship between central angles and inscribed
angles in a circle?
1 ENGAGE
COMMON
CORE
CC.9-12.G.C.2
Introducing Angles and Arcs
In order to begin working with circles, it is helpful to introduce some vocabulary.
DE and EF are
chords.
∠DCF is a central
angle.
∠DEF is an
inscribed angle.
A chord is a segment whose endpoints lie on a circle.
A central angle is an angle whose vertex is the center of a
circle. An inscribed angle is an angle whose vertex lies on a
circle and whose sides contain chords of the circle.
An arc is a continuous portion of a circle consisting of two points on the circle, called
the endpoints of the arc, and all the points of the circle between them. The table summarizes arc
measurement and arc notation.
Arc
Measure/Notation
A minor arc is an arc whose
points are on or in the
interior of a central angle.
The measure of a minor arc
is the measure of its central
angle.
= m∠DCF
A major arc is an arc whose
points are on or in the
exterior of a central angle.
The measure of a major arc
is 360° minus the measure of
its central angle.
= 360° – m∠DCF
Figure
A semicircle is an arc whose The measure of a semicircle
endpoints are the endpoints is 180°.
of a diameter.
= 180°
1
REFLECT

1a. Explain how

1b. The minute hand of a clock sweeps out an arc as the time progresses from 12:05 to 12:20. What is
the measure of the arc? Explain.
compares to
.

Two arcs of a circle are adjacent arcs if they share an endpoint. The following postulate states that
you can add the measures of adjacent arcs. Arc Addition
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