1. Geometry
9.5 Inscribed Angles

2. 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

3. Inscribed Angle Theorem
The measure of the inscribed angle is half the measure of its central angle (and therefore half the intercepted arc).
30o
80o
160o
60o
60o
160o

4. A Very Similar Theorem
The measure of the angle created by a chord and a tangent equals half the intercepted arc.
tangent
50o
tangent
35o
chord
chord
100o
70o

5. Corollary
If two inscribed angles intercept the same arc, then the angles are congruent. ~
sf = giants ~
x = y y x
giants sf

6. Corollary
If an inscribed angle intercepts a semicircle, then it is a right angle.
Why?
180o
diameter
diameter
90o

7. Corollary
If a quadrilateral is inscribed in a circle, then opposite angles are supplementary.
70o
85o
supplementary
supplementary
95o
110o

8. Solve for the variables.1. 2. 3.
140o O y
20
110 x O y
20 x
60
140 O y x
90o
100o
75o
20o
150o
Semicircle
120o
40o
Angle x and the 20o angle
intercept the same arc.
x = 40o
x = 60o
y = 75o
x = 20o
y = 50o
y = 90o

9. Solve for the variables.
x and y both intercept a semicircle. 4. 5. 6.
Inscribed Quadrilateral O x
80 y O x y z O x z y
82
x = 98o
Part of semicircle
100o
supplementary
180o
100o
y + 82o + z = 180o
x = 40o
y + z = 98o
x = 90o
The red and orange arcs are congruent
(they have congruent chords).
y = 50o
y = 90o
Thus, y and z are congruent angles
(they intercept the red and orange arcs).
z = 90o
y = 49o
z = 49o

10. Find x and the measure of angle D.
Inscribed Quadrilateral A B C D 4x
15x
50 x2 7. 8. A B C D 5x 8x x2
supplementary
If x is negative, this angle
would have a negative value.
100o
If x is negative, this angle
would have a negative value.
X2 + 15x = 100
X2 + 8x = 180
X2 + 15x = 0
X2 + 8x = 0
( )( ) = 0
x x -
( )( ) = 0
x x -
x + 20 = 0 and x – 5 = 0
x + 18 = 0 and x – 10 = 0
x = -20 and x = 5
x = -18 and x = 10

11. HW