Inscribed Angles

Section 10.5

Essential Questions

1. How to use inscribed angles to solve problems in geometry

2. How to use inscribed angles to solve real-life problems

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

inscribed angle

intercepted arc

Measure of an Inscribed Angle Theorem (10.9)

If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

C

A

D BmADB =

1

2mAB

Example 1

Find the measure of the blue arc or angle.

RS

QT

a.

mQTS = 2(90 ) = 180

b.80

E

FG

mEFG = 1

2(80 ) = 40

Congruent Inscribed Angles Theorem (10.10)If two inscribed angles of a circle intercept

the same arc, then the angles are congruent.

A

CB

D

C D

Example 2

It is given that mE = 75 . What is mF?

D

E

HF

Since E and F both interceptthe same arc, we know that theangles must be congruent.

mF = 75

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(10.11)

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

C

BB is a right angle if and only if ACis a diameter of the circle.

Inscribed Quadrilateral Theorem(10.12)

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

C

E

F

DG

D, E, F, and G lie on some circle, C if and only if mD + mF = 180 and mE + mG = 180 .

Example 3

Find the value of each variable.

2x

Q

A

B

C

a.

2x = 90

x = 45

b. z

y

80

120

D

E

F

G

mD + mF = 180

z + 80 = 180

z = 100

mG + mE = 180

y + 120 = 180

y = 60

Assignment:P. 508/ 1- 14

HW: p. 508/ 15- 33 all