Geometry

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
x
y
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.
140o
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.
Inscribed Quadrilateral
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
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 - 100 0
X2 8x - 180 0
( )( ) 0
x x -
20 5
( )( ) 0
x x -
18 10
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