arcs and chords lesson 10.2 california state standards 4: prove theorems involving congruence and...
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Arcs and Chords lesson 10.2
California State Standards4: Prove theorems involving congruence and similarity7: Prove/use theorems involving circles.16: Find probability given graph or table
21: Prove/solve relationships with circles.
definitionsCentral Angle
An angle whose vertex is the center of a circle.
Minor ArcA section of the circle cut by a central angle that
measures less than 180o.
Major ArcA section of the circle that measures more than 180o.
SemicircleAn arc with endpoints coinciding with the endpoints
of a diameter
definitions
C
A
B
definitions
C
360mADB m ACB
mAB m ACB
definitions
Measure of a minor arcEquals the measure of its central angle.
Measure of a major arcEquals the difference between 360o and
the measure of its associated minor arc.
C
A
BD
m ACB m ECD
AB DE
definitions
Congruent ArcsArcs with the same measure within the same circle or congruent circles.
C
A
BD
E
because
mQRmPQ
postulate
Arc Addition PostulateThe measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
C
PQ
R
PQR
180omEHI
328omHIE
example
C
H
I
E
Find the measure of each arc.
148o
32omEH
mSE
mSET
mST
example
C
D
T
S
Find the measure of each arc.
82o
142o
218o
118o
60o
100o
E
mDE
mAB
example
C
B
D
A
E
60o
Find the measure of each arc.
60o
60o
yes
Is AD BE ?
mBD
mAE
No
Is AE BD ?
examples
C
B
D
A
E
210o
Find the measure of each arc.
150o
150o
The circles are not
congruent
mBD
mAE
No
Is ? AD BE
examples
C
B
D
A
E
210o
Find the measure of each arc.
150o
150o
The circles are not
congruent homework
if AB PQ
theoremCongruent Arcs and Chords
Two minor arcs are congruent if and only iftheir corresponding chords are congruent.
C B
Q
P
A
then AB PQ
if AB PQ
theoremCongruent Arcs and Chords
Two minor arcs are congruent if and only iftheir corresponding chords are congruent.
C B
Q
P
A
then AB PQ
theoremDiameter-Chord
A diameter that is perpendicular to a chordbisects the chord and its arc.
C
B
A
D
E
AE BE and AD ED
is a diameterDF
is the bisector of DF AB
theoremDiameter-Chord Converse
If one chord is the perpendicular bisector of another chord,
then the first chord is a diameter.
C
B
A
D
E
F
If , then
PQ ABCR CD
theoremsCongruent Chords
Two chords are congruent if and only if they are equidistant from the center.
C B
Q
P
A
R
D
If , then
CR CDPQ AB
mBD
3 11 2 48x x 11 48x
37x
Find mBD
example
C
B
DE
(2x + 48)o(3x + 11)o
2(37) 48122
mBE 180 10476
mDE 26 78104
4 78x x 3 78x
26x
examples
C
B
D
E
4xo(x + 78)o
Find mBE
6FE example
B
C
E
A
D
G
F
AB = 12 DE = 12 CE = 7
Find CG 6
7
2 2 27 6 CF 249 36 CF
213 CF13 CF
CF CG13 CG13
LOGICAL REASONING What can you conclude about the diagram?State a postulate or theorem that justifies your answer.
MEASURING ARCS AND CHORDS Find the measure of the red arc or chord in circle A. Explain your reasoning.
MEASURING ARCS AND CHORDS Find the value of x in circle C. Explain yourreasoning.
TIME ZONE WHEEL In Exercises 49–51, use the following information.The time zone wheel shown at the right consists of two concentric circular pieces of cardboard fastened at the center so the smaller wheel can rotate. To find the time in Tashkent when it is 4 P.M. in San Francisco, you rotate the small wheel until 4 P.M. and San Francisco line up as shown. Then look at Tashkent to see that it is 6 A.M. there. The arcs between cities are congruent.
49. What is the arc measure for each time zone on the wheel?
50. What is the measure of the minor arc from the Tokyo zone to the Anchorage zone?
51. If two cities differ by 180° on the wheel, then it is 3:00 P.M. in one city if and only if it is ____ in the other city.
PROVING THEOREM 10.4 In Exercises 56 and 57, you will prove Theorem 10.4 for the case in which the two chords are in the same circle.
1. AB CD2. APB DPC
3. AP DP BP CP 4. APB DPC 5. AB DC
1. given
2. def arc measure
3. def radius
4. SAS
5. CPCTC
PROVING THEOREM 10.4 In Exercises 56 and 57, you will prove Theorem 10.4 for the case in which the two chords are in the same circle.
PROVING THEOREM 10.7 Write a proof.