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Lesson 6.3 Inscribed Angles and their Intercepted Arcs

Objectives:

Using Inscribed Angles

Using Properties of Inscribed Angles.

Homework: Lesson 6.3/ 1-12Friday-Chapter 6 Quiz 2 on 6.1-6.3

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Using Inscribed Angles

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

Inscribed Angles & Intercepted Arcs

D

B A

C

∠ABC is an inscribed angle

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Using Inscribed Angles

Measure of an Inscribed Angle

2

1

50°

100°

B

AC

50°

100°

B

AC

x°

2x°

B

AC

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Using Inscribed Angles

Example 1:

63

Find the m and mPAQ .PQ

PQ = 2 * m PBQ = 2 * 63 = 126˚

PQ

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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˚

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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

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Using Inscribed Angles

Example 3:

70E D

A

FEDFmFind

14070*2 EFm

EDFm =360 – 140 = 220˚

m = 82˚

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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

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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

A

B

F

E

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Using Properties of Inscribed Angles

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

Cyclic Quadrilateral Conjecture

m A + m C = 180°∠ ∠

m B + m D = 180°∠ ∠

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Using Properties of Inscribed Angles

Find the measure ofGDE

Opposite angles of an inscribed quadrilateral are supplementary

Intercepted arc of an inscribed angles = 2* angle measure

Example 5:

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Find m A and ∠ m B∠ Opposite angles of an inscribed quadrilateral are supplementary

m A + 60° = 180°∠

m A = 120°∠

m B + 140° = 180°∠

m B = 40°∠

Example 6:

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3x°

(y + 5)°

(2y - 3)°

Using Properties of Inscribed Angles

Find x and y Opposite angles of an inscribed quadrilateral are supplementary

Example 7:

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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

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Using Properties of Inscribed Angles

A triangle inscribed in a circle is a right triangle if and only if the diameter is the

hypotenuse         

Angles inscribed in a Semi-circle Conjecture

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

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Find x.

3x°E

D

A

B

C

F

Angles inscribed in a semi-circle are right angles

Example 8:

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Using Inscribed Angles

E DA

B

FFind mFDE

Example 9:146°

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Using Properties of Inscribed Angles

Parallel (Secant) Lines Intercepted Arcs Conjecture

Parallel (secant) lines intercept congruent arcs.

A

B

X

Y

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Using Properties of Inscribed Angles

Find x.

x122˚

189˚

360 – 189 – 122 = 49˚

x = 49/2 = 24.5˚

x

Example 10:

Tangent/Chord Conjecture

BDC2

1C mm

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

B

C

D

BD

C

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Example 11:

Using Tangent/Chord Conjecture

35o

xo

yoQ

L

K

J

Find x and y.

90 QJL m

90o

55x

55o

125180 x

35y

Triangle sum

Example 12:

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Homework:

Lesson 6.3/ 1-12

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