Download 12-3 - Ithaca Public Schools

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Lesson 12-3
Inscribed Angles
Lesson Objectives
1 Find the measure of an inscribed
angle
2 Find the measure of an angle formed
by a tangent and a chord
NAEP 2005 Strand: Geometry
Topic: Relationships Among Geometric Figures
Local Standards: ____________________________________
Vocabulary and Key Concepts.
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Theorem 12-9: Inscribed Angle Theorem
A
The measure of an inscribed angle is
B
C
mB Theorem 12-10
The measure of an angle formed by a tangent and a chord is
B
B
D
D
C
C
Corollaries to the Inscribed Angle Theorem
1. Two inscribed angles that intercept the same arc are
2. An angle inscribed in a semicircle is a
.
angle.
3. The opposite angles of a quadrilateral inscribed in a circle are
An inscribed angle has
.
A
B
C
An intercepted arc is
234
Geometry Lesson 12-3
Daily Notetaking Guide
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mC Name_____________________________________ Class____________________________ Date ________________
Examples.
x
1
mDEF
70°
E
1 Using the Inscribed Angle Theorem Find the values of x and y.
F
80°
Inscribed Angle Theorem
C
x
(m
)
m
D y°
Arc Addition Postulate
x°
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x
(
)
90°
Substitute.
x
G
Simplify.
0
1
Because EFG is the intercepted arc of D, you need to find mFG in order
1
to find mEFG . The arc measure of a circle is 360, so
0
.
mFG 360 y
1
mEFG
y
(m
y
(
Inscribed Angle Theorem
m
)
Arc Addition Postulate
)
Substitute.
y
Simplify.
2 Using Corollaries to Find Angle Measures Find the values of a and b.
By Corollary 2 to the Inscribed Angle Theorem, an angle inscribed in a
semicircle is a right angle, so a .
b°
32°
O
The sum of the measures of the three angles of the triangle inscribed in
O is
. Therefore, the angle whose intercepted arc has
a°
measure b must have measure 180 Because the inscribed angle has
the measure of the intercepted
arc, the intercepted arc has
angle, so b 2(
Daily Notetaking Guide
)
or
.
the measure of the inscribed
.
Geometry Lesson 12-3
235
Name_____________________________________ Class____________________________ Date________________
B
3 *Using
) Theorem 12-10 RS and TU are diameters of circle A.
T
RB is tangent to A at point R. Find mBRT and mTRS.
0
mRT m
mBRT Theorem 12-10
m
N
Arc Addition Postulate
(
mBRT )
Substitute
and
126°
A
S
for mURT
for mUR.
U
Simplify.
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mBRT R
0
mRT
Use the properties of tangents to find mTRS.
A tangent is
mBRS the radius of a circle at its point of tangency.
mBRS m
to
m
mTRS
mTRS
Angle Addition Postulate
Substitute.
Simplify.
Quick Check.
2. For the diagram at the right, find the measure of each numbered angle.
4
1 60°
2
80°
3
3. In Example 3, describe two ways to find mNRS using Theorem 12-10.
236
Geometry Lesson 12-3
Daily Notetaking Guide
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
1. In Example 1, find mDEF.