Download 10.2Honors.notebook 1 March 25, 2012

10.2Honors.notebook
March 25, 2012
Arcs
Angles and Arcs
Section 10.2: Arcs and Chords
1. I can define arcs of a circle and apply their properties
to solve problems.
2. I define chords of circles and apply their properties to
solve problems.
A
B
D
• The sum of the measures
of the central angles of a
circle is 360 degrees.
• A central angle separates
the circle into parts, each
of which is an arc.
• Arc - a part or portion of a
circle that is defined by two
endpoints.
c
Textbook HW Section 10.2
Arcs of a Circle: Graphic Organizer
Type of Arc:
Minor Arc
A
Example:
Major Arc
Semicircle
D
J
1. (HW# 12) What of arc is arc PQ?
a. minor arc
b. major arc
C
60
B
M
G
F
Named:
Arc Degree Measure Equals
Usually by the letters of the two endpoints. Ex: By the letters of the two endpoints and another point on the arc. Ex:
N
K
L
By the letters of the two endpoints and another point on the arc. Ex: The measure of the 360 minus the measure 360 ÷2 or 180 degrees central angle. Less of the minor arc. than 180 degrees.
Greater than 180 degrees. A
P
C
central angle
B
major arc - an arc
with a measure of
more than 180
degrees.
minor arc - an arc with a
measure of less than 180
degrees.
The measure of each arc is related to
the measure of its central angle.
Postulate 26:
c. semicircle
2. (HW#14) What type of arc is arc PQT?
a. minor arc
b. major arc
c. semicircle
E
110
Arcs are usually denoted by this symbol_______.
AB
Central angle - an angle that intersects a circle in two
points and has its vertex at the center. The sides of
the central angle contain two radii of the circle.
3. (HW#16) What type of arc is arc TUQ?
a. minor arc
b. major arc
c. semicircle
4. (HW#20) What is the measure of arc KL?
5. (HW#22) What is the measure of arc LNK?
Two arcs of the same circle are adjacent if they intersect at exactly
one point.
A
C
Arc Addition Postulate: The
B
measure of an arc formed by two
adjacent arcs is the sum of the
measure of the two arcs.
__________________________
6. (HW#24) What is the measure of arc NJK?
7. On a scale of 1 ­ 5, with one being the highest, how confident are you
with the statement: I can identify arcs of circles and their measures? 1
10.2Honors.notebook
March 25, 2012
Measures of Arcs using Arc Addition Postulate:
Find the measure of each arc:
C
A
Arc AB ____________
40
E
Arc ABD ___________
80
110
B
Arc AD _____________
Congruent Arcs
Remember...
how do we
determine if two
circles are
congruent?
congruent arcs - two arcs of the
same circle or of congruent circles
are congruent arcs if their
measures are the same.
• Two minor arcs, of the same
circle or of congruent circles, are
congruent if their central angles
are congruent.
60
60
3.
(X+40)
C
A
Find the measure
of arc AD.
B
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. A
C
B
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. F
E
G
D
Z
Theorem 10.6 W
D
Theorem 10.5 2.
Example Theorem
Theorem 10.4
45
C
Y
2X
45
Are these arcs
congruent?
65
Example: Theorem 10.4
D
B
X
D
Theorems About Chords and Circles
A
1.
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. M
J
K
L
Homework Problems:
Page 608: 32 - 46 even
Theorem 10.7
In the same circle or in congruent circles, two chords are congruent if
and only if they are equidistant from the center.
C
G
E
A
F
D
B
Example:
CD = 40
ED = 25
Find EF =
2