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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