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Segments and Rays. Lesson 1-2. 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. - PowerPoint PPT Presentation
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Lesson 1-2: Segments and Rays 1
Lesson 1-2
Segments
and Rays
Lesson 1-2: Segments and Rays 2
Postulates
Definition: An assumption that needs no explanation.
Examples:
• Through any two points there is exactly one line.
• Through any three points, there is exactly one plane.
• A line contains at least two points.
• A plane contains at least three points.
Lesson 1-2: Segments and Rays 3
Postulates
• 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.
Examples:
Lesson 1-2: Segments and Rays 4
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
-5 5
SRQPOLKJIHG M N
PK = | 3 - -2 | = 5 Remember : Distance is always positive
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 │
Lesson 1-2: Segments and Rays 6
Between
Definition: X is between A and B if AX + XB = AB.
A BX
AX + XB = AB AX + XB > AB
A BX
Lesson 1-2: Segments and Rays 7
Segment
Part of a line that consists of two points called the endpoints and all points between them.
How to sketch:
How to name:
Definition:
AB
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 8
The Segment Addition Postulate
AB
C
If C is between A and B, then AC + CB = AB.Postulate:
Example: If AC = x , CB = 2x and AB = 12, then, find x, AC and CB.
AC + CB = AB
x + 2x = 12
3x = 12
x = 4
2xx
12
x = 4AC = 4CB = 8
Step 1: Draw a figure
Step 2: Label fig. with given info.
Step 3: Write an equation
Step 4: Solve and find all the answers
Lesson 1-2: Segments and Rays 9
Congruent Segments
Definition:
If numbers are equal the objects are congruent.
AB: the segment AB ( an object )
AB: the distance from A to B ( a number )
AB
D
C
Congruent segments can be marked with dashes.
Correct notation:
Incorrect notation:
AB = CD AB CD
AB = CDAB CD
Segments with equal lengths. (congruent symbol: )
Lesson 1-2: Segments and Rays 10
Midpoint
a b
2
1 1( , )x y 2 2( , )x y
A point that divides a segment into two congruent segments
Definition:
EDFIf DE EF , then E is the midpoint of DF.
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 .1 2 1 2,2 2
x x y y
Formulas:
Lesson 1-2: Segments and Rays 11
Midpoint on Number Line - Example
Find the coordinate of the midpoint of the segment PK.
-5 5
SRQPOLKJIHG M N
a b 3 ( 2) 10.5
2 2 2
Now find the midpoint on the number line.
Lesson 1-2: Segments and Rays 12
Segment BisectorAny segment, line or plane that divides a segment into two congruent parts is called segment bisector.
Definition:
B
E
D
FA
BE
D
FA
E
D
A F
B
AB bisects DF. AB bisects DF.
AB bisects DF.Plane M bisects DF.
Lesson 1-2: Segments and Rays 13
Ray
Definition:
( the symbol RA is read as “ray RA” )
How to sketch:
How to name:
R
A R A Y
RA ( not AR ) RA or RY ( not RAY )
RA : RA and all points Y such that A is between R and Y.
Lesson 1-2: Segments and Rays 14
Opposite Rays Definition:
( Opposite rays must have the same “endpoint” )
AX Y
D ED E
opposite rays not opposite rays
DE and ED are not opposite rays.
If A is between X and Y, AX and AY are opposite rays.