Lesson 1-2: Segments and Rays 1

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

Examples :
• If two planes intersect, then the intersecting is a line.
• If two points lie in a plane, then the line containing the two points lie in the same plane.
Lesson 1-2: Segments and Rays 3

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 4

G
Find the distance between P and K .
H I J K L
M
N O P Q R
-5 5
Note: 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 │
S
PK = | 3 - 2 | = 5 Remember : Distance is always positive
Lesson 1-2: Segments and Rays 5

Definition:
A X
X is between A and B if AX + XB = AB.
X
B A B
Lesson 1-2: Segments and Rays 6

Definition: Part of a line that consists of two points called the endpoints and all points between them.
A
How to sketch: B
How to name:
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 7

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
A x
C
2x
Step 1: Draw a figure
12
Step 2: Label fig. with given info.
AC + CB = AB
Step 3: Write an equation x + 2x = 12
Step 4: Solve and find all the answers
3x = 12 x = 4
Lesson 1-2: Segments and Rays x = 4
AC = 4
CB = 8
8

Definition
:
Segments with equal lengths.

Congruent segments can be marked with dashes.
A
B
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 :

Incorrect notation:

Lesson 1-2: Segments and Rays 9

Definition: A point that divides a segment into two congruent segments
If DE

EF , then E is the midpoint of DF.
D
Formulas:
E
On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is .
2
In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and
2
) x
1
 is .
2 x
2 , y
1

2 y
2
Lesson 1-2: Segments and Rays 10
F

Find the coordinate of the midpoint of the segment PK .
G H I J K L
M
N O P Q R S
-5 5 a
 b

 
1
2 2
 
2
0.5
Now find the midpoint on the number line.
Lesson 1-2: Segments and Rays 11

D
Definition:
A
E
Any segment, line or plane that divides a segment into two
F congruent parts is called segment bisector.
A
B
E
F
AB bisects DF.
D
B
AB bisects DF.
F
A
E
Plane M bisects DF.
D B
AB bisects DF.
Lesson 1-2: Segments and Rays 12

Definition:
RA : RA and all points Y such that
A is between R and Y.
A
How to sketch:
R A
R
Y
How to name: RA ( not AR )
( the symbol RA is read as “ray RA” )
RA or RY ( not RAY )
Lesson 1-2: Segments and Rays 13

Definition :
If A is between X and Y, AX and AY are opposite rays.
( Opposite rays must have the same “endpoint” )
opposite rays not opposite rays
not
Lesson 1-2: Segments and Rays 14