1. Lesson 1-2: Segments and Rays

2. Lesson 1-2: Segments and Rays
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

3. Lesson 1-2: Segments and Rays
Postulates
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

4. Lesson 1-2: Segments and Rays
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

5. Ruler Postulate : Example
Find the distance between P and K.
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 │
PK =
| | = 5
Remember : Distance is always positive
Lesson 1-2: Segments and Rays

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

7. Lesson 1-2: Segments and Rays
Definition:
Part of a line that consists of two points called the endpoints and all points between them.
How to sketch:
How to name:
AB (without a symbol) means the length of the segment or the distance between points A and B.
Lesson 1-2: Segments and Rays

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

9. Lesson 1-2: Segments and Rays
Congruent Segments
Definition:
Segments with equal lengths. (congruent symbol: )
Congruent segments can be marked with dashes.
If numbers are equal the objects are congruent.
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

10. Lesson 1-2: Segments and Rays
Midpoint
Definition:
A point that divides a segment into
two congruent segments
Formulas:
On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is
In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and is
Lesson 1-2: Segments and Rays

11. Midpoint on Number Line - Example
Find the coordinate of the midpoint of the segment PK.
Now find the midpoint on the number line.
Lesson 1-2: Segments and Rays

12. Lesson 1-2: Segments and Rays
Segment Bisector
Definition:
Any segment, line or plane that divides a segment into two congruent parts is called segment bisector.
Lesson 1-2: Segments and Rays

13. Lesson 1-2: Segments and Rays
Definition:
RA : RA and all points Y such that
A is between R and Y.
How to sketch:
How to name:
( the symbol RA is read as “ray RA” )
Lesson 1-2: Segments and Rays

14. Lesson 1-2: Segments and Rays
Opposite Rays
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
Lesson 1-2: Segments and Rays