Section 9.5

INSCRIBED ANGLES

Inscribed Angle• What does inscribe mean?• An inscribed angle is an angle whose vertex is on a

circle and whose sides contain chords of the circle.

An inscribed angle

is equal to half of

its intercepted arc

1 11 ( )2

m arc chord

cho

rd

Central vs. Inscribed

• How are central angles related to the arcs they make?

1

2

Central Angles are = to the arcs

Inscribed Angles are = ½ of the arcschord

cho

rd radius

rad

ius

1) Find the values of x and y.

y

110

x

20oo

o

o

x is opposite of the inscribed angle of 20o

x = 40o

x = 40o

y is the inscribed angle of the arc including 110o and 40o

y = ½(150o) = 75o

y = 75o

2) Find the values of x, y, and z.

zx

y

60

o

o

o

o y is a central angle of arc of 60o

y = 60o

y = 60o

x is an inscribed angle of arc of 60o

x = ½(60o) = 30o

x = 30o

zo

360 – 60 = 300 = 2z

z = 150o

1) Draw a circle.

2) Draw a diameter and label the points A & B.

3) Label another point Q on the circle.

4) Highlight the semicircle around these letters.

5) Draw ∠AQB.

6) What should be true about that angle ALWAYS?Q

Semicircles

A

B

An angle inscribed in a semicircle is a

right angle.

20

y

xCenter

oo

o

3) Find the values of x and y.A diameter is pictured.

y = 90o because it is an inscribed angle of a semicircle

40o

x = 20o because it is an inscribed angle of this arc

Quads and Circles1) Draw a circle.

2) Inscribe a quadrilateral into a circle.

3) Label the points of the quadrilateral DOGS.

4) What do you think is true about the opposite angles?

If a quad. is inscribed into a circle, then its opp. angles are supplementary.

D O

GS

Last one!

• The measure of an angle formed by a

chord and a tangent is equal to half

the measure of the intercepted arc.

140

1

What is the

m∠1? chor

d

tangent

m∠1 = ½(140o)

= 70o

x

82y

z

o

o

o

o

4) Find the values of x, y, and z.

x is opposite of 82o so they are supplementary.

x = 98o

y and z are congruent angles because their chords are congruent.

164o

360 – 164 = 196 196/2 = 98o

98o

98o

y and z are inscribed so 98/2 = 49o

Practice Problems

• Page 353• Classroom Exercises• #4 - 9

• DRAW ALL DIAGRAMS