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Geometry
Segments
and Rays
Lesson: Segments and Rays
1
Postulates
Definition: An assumption that needs no explanation.
Examples:
• Through any two points there is
exactly one line.
• A line contains at least two points.
• Through any three points, there is
exactly one plane.
• A plane contains at least three points.
Lesson 1-2: Segments and Rays
2
The Ruler Postulate
The Ruler Postulate: Points on a line can be paired with the real
numbers in such a way that:
• Any two chosen points can be paired with 0 and 1.
• The distance between any two points on a number line is the
absolute value of the difference of the real numbers corresponding
to the points.
Formula: Take the absolute value of the difference of the two
coordinates a and b: │a – b │
Lesson 1-2: Segments and Rays
3
Ruler Postulate : Example
Find the distance between P and K.
G
H
I
J
K
L
M
N
O
P
-5
Note:
Q
R
S
5
The coordinates are the numbers on the ruler or number line!
The capital letters are the names of the points.
Therefore, the coordinates of points P and K are 3 and -2 respectively.
Substituting the coordinates in the formula │a – b │
PK = | 3 - -2 | = 5
Remember : Distance is always positive
Lesson 1-2: Segments and Rays
4
Measuring Segment Lengths




What is ST?
What is SV?
What is UV?
What is TV?
Measuring Segment Lengths




ST = | -4 – 8 | = | -12| = 12
SV = |-4 – 14 | = | -18| = 18
UV = | 10 – 14| = | -4| = 4
TV = |8 – 14| = | -6 | = 6
Postulate: Segment Addition Postulate
 If three points A, B, and C are collinear and B is
between A and C, then AB + BC = AC.
Between
Definition:
X is between A and B if AX + XB = AB.
X
A
X
B
AX + XB = AB
A
B
AX + XB > AB
Lesson 1-2: Segments and Rays
8
Segment
Definition: Part of a line that consists of two points called the
endpoints and all points between them.
How to sketch:
How to name:
A
B
AB or BA
The symbol AB is read as "segment AB".
AB (without a symbol) means the length of
the segment or the distance between points
A and B.
Lesson 1-2: Segments and Rays
9
The Segment Addition Postulate
Postulate: If C is between A and B, then AC + CB = AB.
Example: If AC = x , CB = 2x and AB = 12, then, find x, AC
and CB.
B
2x
A x C
Step 1: Draw a figure
12
Step 2: Label fig. with given info.
AC + CB = AB
x + 2x = 12
Step 3: Write an equation
Step 4: Solve and find all the answers
3x = 12
x = 4
Lesson 1-2: Segments and Rays
x = 4
AC = 4
CB = 8
10
Congruent Segments
Definition: Segments with equal lengths. (congruent symbol:
)
B
Congruent segments can be marked with dashes.
A
If numbers are equal the objects are congruent.
C
D
AB: the segment AB ( an object )
AB: the distance from A to B ( a number )
Correct notation:
AB = CD
AB  CD
Incorrect notation:
AB  CD
AB = CD
Lesson 1-2: Segments and Rays

11
Midpoint
Definition: A point that divides a segment into
two congruent segments
If DE  EF , then E is the midpoint of DF.
Formulas:
F
E
D
On a number line, the coordinate of the midpoint of a segment
whose endpoints have coordinates a and b is a  b .
2
In a coordinate plane, the coordinates of the midpoint of a
segment whose endpoints have coordinates ( x1 , y1 ) and ( x2 , y2 )
is
 x1  x2 y1  y2 
,


2 
 2
.
Lesson 1-2: Segments and Rays
12
Midpoint on Number Line - Example
Find the coordinate of the midpoint of the segment PK.
G
H
I
J
K
L
M
N
O
P
Q
-5
R
S
5
a  b 3  (2) 1

  0.5
2
2
2
Now find the midpoint on the number line.
Lesson 1-2: Segments and Rays
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Segment Bisector
Definition: Any segment, line or plane that divides a segment into two
congruent parts is called segment bisector.
A
F
A
B
E
AB bisects DF.
B
D
F
E
D
F
A
E
D
AB bisects DF.
Plane M bisects DF.
B
AB bisects DF.
Lesson 1-2: Segments and Rays
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