Transcript Geometry - San Ramon Valley High School

Geometry
9.3 Arcs and Central Angles
Central Angles

An angle with the vertex at the center
of the circle.
A
AQX, AQB, and YQX are examples
of central angles.
7
X
Q
Y
B
7
7
Arc

An unbroken part of the circle.
A
AB
X
Q
B
Y
XBA
Measures of an arc
Semicircle
Minor Arcs
Has a measure of 180 degrees.
Needs three letters in its symbol.
Has a measure between 0 and 180 degrees.
Needs only two letters in its symbol.
A
A
AX
X
X
Q
B
Y
Major Arcs
Q
Has a measure between 180 and 360 degrees.Y
Needs three letters in its symbol.
A
The measure of a minor arc is equal
to the measure of its central angle.
X
Q
Y
B
AXY
B
XBY
Using the letters shown in the diagram, name:
X
W
Q
1. four central angles
7
XQY
YQZ
7
7
WQX
XQZ
Z
7
2. two semicircles
WXY
XYZ
3. four minor arcs
WX
YX
ZY
WZ
4. four major arcs
WXZ
Y
WZX
Are these the same?
YZX
ZXY
Adjacent Arcs

Arcs with exactly one point in
common.
J
IJ and JK are adjacent arcs.
I
K
Are arcs that overlap adjacent?
No, because they would have more than one common point.
Arc Addition Postulate

The measure of the arc formed by two adjacent arcs is the
sum of the measures of these two arcs.
B
mBC + mCD = mBCD
C
A
Find the mistake on your handout.
D
Minor arc only needs two letters.
Find each measure.
45
5. PCQ
60o
8. SQ
120o
11. SPQ
240o
6. ST
45o
9. SCQ
120o
12. PT
135o
15. TSQ
14. SPT
97.5o
360 – 45 = 315o
7. SQP
180o
10. SCP
180o
13. TCP
135o
T
S
135o
C
120o
P
Q
60
Find the measure of each numbered angle. O is the center of the circle.
17.
18.
1
1
60o
2
O
2
O
O
1
m 1 = 180o – m 2
m 2 = 180o – m 1
7
120o
40
1
7
O
19.
7
240
7
16.
40o
140o
Congruent Arcs

Arcs in the same circle or congruent circles that
have equal measures are congruent.
A
T
R
RY = QA ≠ SP
Q
X
B
S
P
Y
C
XY ~
= AB
but neither arc is congruent to ST because circle P is not congruent
to the other two circles.
Theorem
In the same circle or in congruent circles, two
minor arcs are congruent if and only if their central
angles are congruent.
J
M
2
~ LM.
If m 1 = m 2, then JK =
7
7
~ LM, then m 1 = m 2.
If JK =
7
7

1
L
K

The figure shows two concentric circles with center N.
A
Classify each statement as true of false
45
B
V
W
AB  VW
20. mBC  45
True
21.
22. mDNC  90
23. mXY  45
True
24.
VW  WX
True
N
False
X
E
False
Z
Y
D
25.
AED  VZY
False
True/False: mAB = mVW
True
C
HW

P. 341-342 WE 1-11, 16-18
for 17-18 see example P. 340