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Geometry
Name:
12-3: Inscribed Angles
Date:
Finding the Measure of an Inscribed Angle
Period:
Corollaries to the Inscribed Angle Theorem
At the right, the vertex of C
is on O, and the sides of C
are chords of the circle, C is
an inscribed angle. AB is the
intercepted arc of C.
1. Two inscribed angles that intercept the same
arc are congruent.
2. An angle inscribed in a semicircle is a right
angle.
3. The opposite angles of a quadrilateral
inscribed in a cycle are supplementary.
Theorem 12-9 describes the relationship between an
inscribed angle and it intercepted arc.
Theorem 12-9
Inscribed Angle Theorem
Example 2 – Using Corollaries to Find Angle
Measure
Find the angle of the numbered angle.
The measure of an inscribed
angle is half the measure of its
intercepted arc.
a.
b.
mÐB = 12 mAC
To prove Theorem 12-9, there are three cases to
consider.
The Angle Formed by a Tangent and a Chord
I: The center is on a II: The center is
side of the angle.
inside the angle.
III: The center is
outside the angle.
A proof of Case I is below. You will prove Cases II
and III in the More Practice tomorrow.
Example 1 – Using the Inscribed Angle Theorem
Find the values of a and b.
In the diagram below, B and C are fixed points, and
point A moves along the circle. From the Inscribed
Angle Theorem, you know that as A moves, mA
remains the same and is 12 mBC . As the last diagram
suggests, that is also true when A and C coincide.
(Coincide means the two points are in the same
location).
Theorem 12-10
Inscribed Angle Theorem
The measure of an angle
formed by a tangent and a
chord is half the measure
of the intercepted arc.
mÐC = 12 mBDC