Download B - Cobb Learning

Session 2
Draw six segments that
pass through every dot in
the figure without taking
your pencil off the paper.
Counterexample
It only takes 1 false example to
show that a conjecture is not true.
Example 4: Find a counterexample for
these statements…
All dogs have spots.
All prime numbers are odd.
Counterexample
Example 5: Find a counterexample:
For all real numbers x, the
expression x2 is greater than or
equal to x.
0.52 = 0.25 and 0.250.5
Point
• Has no size, no dimension
• Is represented by a dot
• Named by using a capital letter
We would call this one “point E.”
Line
• Has one dimension
• Is made up of infinite number of points and is
straight
• Arrows show that the line extends without end
in both directions
• Can be named with a single lowercase cursive
letter OR by any 2 points on the line
• Symbol
Names of these lines:
COLLINEAR Points
lie on the same line
NONCOLLINEAR Points
do NOT lie on the same line
Example
A
D
• Points D, B, & C
are in a straight
line so they are
_______________
C
B
E
• Points A, B, & C
are
________________
Plane
• 2 dimensions
• Extends without end in
all directions
• Takes at least 3
noncollinear pts. to make
a plane
• Named with a single
uppercase script letter
or by 3 noncollinear pts.
Names of these
planes:
M
COPLANAR Points
lie in the same plane
NONCOPLANAR Points
do NOT lie in the same plane
Line
Segment
Is straight and made up of points
•
• Has a definite beginning and definite end
• Name a line segment by using the
endpoints only
• You will always use two letters to name a
segment
• Symbol
Name of these segments:
Name of segment
from 3 to 0.
Ray
Is straight and made up of points
•
• Has a beginning but no end
• Starting pt. of a ray is called the endpoint
• Name a ray by using the endpt. 1st and another
point on the ray
• You will always use two letters to name a ray
• Symbol
Names of these rays:
Postulates
• Postulates are facts about
geometry that are accepted as
true.
Ruler Postulate
Do not copy all of this
• Every point on a
line can be matched
with a coordinate on P
the number line.
• The distance
between two points
is the absolute
value of the
difference of the
coordinates.
Q
Ruler Postulate (in other words)
If P is at 15 and Q
is at 18, the
distance from P to
Q is 3.
P
15
Q
18
The distance from P to Q is written:
PQ
How is this different from PQ?
Ex. 1 Use the number line to find each
measure.
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
B
a) DH
DE F
b) EI
H
I
c) FB
Segment Addition Postulate
If B is between A and C, then
AB + BC = AC.
A
A
B
B
A
C
+
B
C
C
=
Example 2
Points X, Y, and Z are collinear. If XY =
12, YZ = 47, and XZ = 35, determine
which point is between the other two.
35
X
Z
12
X
Y
47
Y
Z
Example 3
If QS = 29 and QT = 52, find ST.
P
Q
R
S
QS + ST = QT
29 + ST = 52
ST = 23
T
Example 4
If FG = 12 and FJ = 47, find GJ.
F
G
H
FG + GJ = FJ
12 + GJ = 47
J
If A(x1, y1) and B(x2, y2) are points on
the coordinate plane, then:
x




x

y

y
2
1
2
1
2
2

5. Find the distance between the points. Round to the nearest tenth.
x
2
 x1    y2  y1 
2
2

eb3  0g b0  4gj
2
b9  16g
25
2
6. Find the distance between the points. Round to the nearest tenth.
x
2
 x1    y2  y1 
2
2

eb4  1g b2  0gj
2
b9  4g
13
2
7. Find the distance between the points. Round to the nearest tenth.
x
2
 x1    y2  y1 
2
2

eb4  1g b1  1gj
2
b25  4g
29
2
Congruent Segments
Two segments are congruent if and
only if they have the same length.
A
B
X
Y
AB  XY
THE SAME INITIAL POINT
VERTEX
TWO DIFFERENT
RAYS
SIDES
A
ABC
CBA
B
1
1
B
C
A ?
MAH 1
OR
HAT
M
H
1
A
OR
2
MAT
2
T
A
B
C
How To Measure An Angle
mRST  mTSP  mRSP
Why can’t you
name any of the
angles S?
T
R
S
P
Example 1
Find m1 if mRSP  78.
m1 + 48 = 78
R
1
T
48
S
P
m1 = 30
Example 2
M
N
42
2
Find m2 if mYJK  160.
m2 + 42 + 104 = 160
104
J
Y
K
m2 + 146 = 160
m2 = 14
Example 3
Find x if mALY  71.
2 x  (5x  8)  71
A
U
2x
(5x  8)
L
Y
7 x + 8 = 71
7x = 63
x = 9
ACUTE ANGLES = Greater than 0 and less than 90
RIGHT ANGLES = Measure exactly 90
OBTUSE ANGLES = Greater than 90 and less than 180
STRAIGHT ANGLES = Measure exactly 180