Download Inscribed Angles

Warm – up
2.
Inscribed Angles
Section 6.4
Standards

MM2G3. Students will understand
the properties of circles.

b. Understand and use properties of
central, inscribed, and related angles.
Essential Questions

What are the important circle
measurements?
Essential Questions


How do I use inscribed angles to solve
problems?
How do I use properties of inscribed
polygons?
Definitions


Inscribed angle – an angle whose vertex is on a
circle and whose sides contain chords of the circle
Intercepted arc – the arc that lies in the interior of an
inscribed angle and has endpoints on the angle
intercepted arc
inscribed angle
Measure of an Inscribed Angle Theorem

If an angle is inscribed in a circle, then its measure
is half the measure of its intercepted arc.
A
mADB =
1
2
C
mAB
D
B
Example 1

a.
Find the measure of the blue arc or angle.
S
R
E
b.
80
F
T
G
Q
mQTS = 2(90) = 180
mEFG =
1
2
(80) = 40
Congruent Inscribed Angles Theorem

If two inscribed angles of a circle intercept
the same arc, then the angles are congruent.
A
B
C
D
C  D
Example 2
It is given that mE = 75. What is mF?
D
Since E and F both intercept
the same arc, we know that the
angles must be congruent.
E
mF = 75
F
H
Definitions


Inscribed polygon – a polygon whose vertices all lie
on a circle.
Circumscribed circle – A circle with an inscribed
polygon.
The polygon is an inscribed polygon and
the circle is a circumscribed circle.
Inscribed Right Triangle Theorem

If a right triangle is inscribed in a circle, then the
hypotenuse is a diameter of the circle. Conversely, if one
side of an inscribed triangle is a diameter of the circle,
then the triangle is a right triangle and the angle opposite
the diameter is the right angle.
A
B is a right angle if and only if AC
is a diameter of the circle.
B
C
Inscribed Quadrilateral Theorem

A quadrilateral can be inscribed in a circle if and only
if its opposite angles are supplementary.
E
F
C
D
G
D, E, F, and G lie on some circle, C if and only if
mD + mF = 180 and mE + mG = 180.
Example 3

Find the value of each variable.
a.
B
Q
A
D
b.
G
y
F
C
x = 45
E
80
2x
2x = 90
z
120
mD + mF = 180
z + 80 = 180
z = 100
mG + mE = 180
y + 120 = 180
y = 60
Practice

Pages 207

2 – 18 even
Homework

Page 209

2 – 26 even