1

Lesson 6.3 Inscribed Angles and their Intercepted Arcs

Goal 1 Using Inscribed Angles

Goal 2 Using Properties of Inscribed Angles.

2

Using Inscribed Angles

An INSCRIBED ANGLE is an angle whose vertex is on the circle and whose sides each contain chords of a circle.

Inscribed Angles & Intercepted Arcs

D

B A

C

3

Using Inscribed Angles

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

m = m arc OR 2 m = m arc

Measure of an Inscribed Angle

21

50°

100°

B

AC

50°

100°

B

AC

50°

100°

B

AC

x°

2x°

B

AC

4

Using Inscribed Angles

Example 1:

63

Find the m and mPAQ .PQ

mPAQ = m PBQmPAQ = 63˚

PQ =2 * m PBQ

= 2 * 63 = 126˚

5

Using Inscribed Angles

Find the measure of each arc or angle.

QSR

Example 2:

Q

R

= ½ 120 = 60˚

= 180˚

= ½(180 – 120)= ½ 60= 30˚

6

Using Inscribed Angles

Inscribed Angles Intercepting Arcs Conjecture

If two inscribed angles intercept the same arc or arcs of equal measure then the inscribed angles have equal measure.

mCAB = mCDB

P

A

BC

D

7

Using Inscribed Angles

Example 3:

70E DA

FEDFmFind

14070*2 EFm

EDFm =360 – 140 = 220˚

m = 82˚

8

Using Properties of Inscribed Angles

Example 4:

41°

60°

P

C

DA

B

Find mCAB and m AD

mCAB = ½

mCAB = 30˚ADm = 2* 41˚ AD

CB

9

Using Properties of Inscribed Angles

Cyclic QuadrilateralA polygon whose vertices lie on the circle, i.e. a quadrilateral inscribed in a circle.

Quadrilateral ABFE is inscribed in Circle O.

O

AB

F

E

10

Using Properties of Inscribed Angles

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

Cyclic Quadrilateral Conjecture

11

Using Properties of Inscribed Angles

A polygon is circumscribed about a circle if and only if each side of the polygon is tangent to the circle.

Circumscribed Polygon

12

Using Inscribed Angles

E DA

B

FExample 5:

Find mEFD

mEFD = ½ 180 = 90˚

13

Using Properties of Inscribed Angles

A triangle inscribed in a circle is a right triangle if and only if one of its sides is a diameter.         

Angles inscribed in a Semi-circle Conjecture

A has its vertex on the circle, and it intercepts half of the circle so thatmA = 90.

14

Using Properties of Inscribed Angles

Find the measure ofGDE

Example 6:

Find x. 3x°E

DA

B

C

F

15

Using Properties of Inscribed Angles

Find x and y

3x°

(y + 5)°

(2y - 3)°

85°80°

y°x°

16

Using Properties of Inscribed Angles

Parallel Lines Intercepted Arcs ConjectureParallel lines intercept congruent arcs.

A

B

X

Y

17

Using Properties of Inscribed Angles

Find x.

x122˚

189˚

360 – 189 – 122 = 49˚

x = 49/2 = 24.5˚

18

Homework:Lesson 6.3/ 1-14