Theorem 12-4: In the same circle (or congruent circles), congruent central angles create congruent intercepted arcs.

then AB CD.

If

mCQD mBQA,

A

B

C D

Q

Theorem 12-5: In the same circle (or congruent circles), congruent central angles create congruent chords.

Then AB CD.

If

mCQD mBQA

A

B

C D

Q

Arcs can be formed by figures other than central angles. Arcs can be formed by

chords, inscribed angles, and tangents. Today we will focus on examining

relationships between chords and their intercepted arcs.

A

B

C Chord AB creates

intercepted minor

arc AB and

intercepted major

arc ACB.

Theorem 12-6: In the same circle (or congruent circles), congruent chords create congruent intercepted arcs.

then AB CD.

If AB CD,

A

B

C D

Example 1 Example 2

AB

CD

mAB = mBD =

mACD =

mBAD =

Given: mAC = 100

mCD = 7575 110

175

250

A

B

C

110 110

250

250

220

mAC = mAB =

mACB =

mABC =

mBAC =

Given: mCB = 140

Theorem 12-8 – A diameter that is perpendicular to a chord, bisects the chord and its intercepted arc.:

B

C

A

D

F

If AB CD,

then CF FD and CB DB.

Also: AD AC.

Example B

AM

Q

LC

Find CA.

Q

LC

CA = 2

Given: AB is a diameter of circle Q; AB = 10, LM = 8.

If mML = 118, find mBL.

mBL = 121

A

Theorem 12-7 – In the same circle (or circles):

1. Chords equally distant from the center are congruent.

2. Congruent chords are equally distant front the center.

B

C

P

RQ

S Remember: To measure distances from a point to a

segment, you have to measure the perpendicular

distance.

1) If AB = BC, then PR QS.

2) If PR QS, then AB = BC.

Example

B

A

C

Q

D

F G

FQ = QG = 9; CB = 24.

Find the length of the radius of circle Q.

B

QF 9

12 BQ = 15

92 + 122 = BQ2

225 = BQ2

Inscribed Angles &

Corollaries

ABC is an inscribed angle of Circle O.

Definition: an Inscribed Angle is an angle with its vertex on the

circle.

A

OC

B

A

B

The measure of the intercepted arc of an inscribed angle is equal to twice

the measure of the inscribed angle.

110°

55°O

C

Theorem 12-11

Corollary 1: Inscribed angles that intercept the same arc are

congruent.A

D

C

B

100°

mABC = 50

mADC = 50

ABC and ADC both

intercept AC.

50°50°

Corollary 2: An angle inscribed inside of a semicircle is a right

angle.A

DCB

70°

35°

110°

55°

mBAC = 90

Here’s why…

Corollary 3: If a quadrilateral is inscribed in a circle, then its

opposite angles are supplementary.

A

D

C

B

mBAC = 76

mACD = 9288°

104°

76°

92°

B

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

A

Theorem 12-12B

E C

D

mBAC = ½ mAB