Download 9.5 Inscribed Angles

Geometry
9.5 Inscribed Angles
Inscribed Angles


The vertex is on the circle
The sides of the angle:
 Are chords of the circle
 Intercept an arc on the circle
Inscribed
angle
Intercepted Arc
Inscribed Angle Theorem

The measure of the inscribed angle is half
the measure of its central angle (and
therefore half the intercepted arc).
30o
60o
60o
80o
160o
160o
A Very Similar Theorem

The measure of the angle created by a
chord and a tangent equals half the
intercepted arc.
tangent
50o
35o
100o
70o
Corollary

If two inscribed angles intercept the same
arc, then the angles are congruent.
~ giants
sf =
x~
=y
y
x
giants
sf
Corollary

If an inscribed angle intercepts a
semicircle, then it is a right angle.
Why?
180o
90o
Corollary

If a quadrilateral is inscribed in a circle,
then opposite angles are supplementary.
70o
85o
95o
110o
Solve for the variables.
1.
2.
90yo
20
75o
y
150o
20o
140
60
110
O
140o
3.
O
20
100o
O
x
y
Semicircle
x
x
40o
x = 40o
y = 75o
Angle x and the 20o angle
intercept the same arc.
x = 20o
y = 90o
x = 60o
y = 50o
120o
Solve for the variables.
4.
5.
x and y both intercept a semicircle. 6.
Inscribed Quadrilateral
80
Part of semicircle
x = 98xo
x
y
y
100o
O
x
O
O
z
100o
180o
x = 40o
y=
50o
y
82
z
y + 82o + z = 180o
y + z = 98o
x = 90o
y = 90o
z = 90o
The red and orange arcs are congruent
(they have congruent chords).
Thus, y and z are congruent angles
(they intercept the red and orange arcs).
y = 49o
z = 49o
Find x and the measure of angle D.
Inscribed Quadrilateral
7.
A
x2
5x
B
8.
4x
A
B
50
15x
If x is negative, this angle
would have a negative value.
8x
D
X2
+ 8x = 180
X2
+ 8x - 180 = 0
If x is negative, this angle
would have a negative value.
( x +18 )(x - 10 ) = 0
x + 18 = 0 and x – 10 = 0
x = -18 and x = 10
C
100o
C
D
x2
X2 + 15x = 100
X2 + 15x - 100 = 0
( x +20 )(x - 5 ) = 0
x + 20 = 0 and x – 5 = 0
x = -20 and x = 5
HW

P. 352-355 CE #1-9 WE #1-9, 19-21