Download Ruler Postulate - Bismarck Public Schools

Lesson 1-2
Use Segments
and Congruence
1
Warm Up Problems
Solve for x.
2x 10  x  4  21
3x 16  4x  8  60
2x  8 17  10  3x
2
Theorem
Definition: A true statement that follows as a result of
other true statements. A statement that can be proven.
We will investigate theorems in greater detail later.
Your book lists all the Theorems (and Postulates )
we will use this year. Page 926
3
Postulate or axiom
Definition: A rule that is accepted without proof.
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.
4
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.
5
POSTULATE 1 The Ruler Postulate
The Ruler Postulate: Points on a line can be matched one to one
with the real numbers.
• The real number that corresponds to a point is the coordinate of
the point.
• The distance between points A and B, written AB, is the absolute
value of the difference of the coordinates of A and B.
•x1 –small numbers are called subscripts (read “x sub one”)
6
Ruler Postulate : Example
Find the distance between P and K.
G
H
I
J
K
L
M
N
O
-5
Note:
P
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
7
Between
Definition:
B is between A and C if AB + BC = AC.
X
A
AB + BC = AC
B
AX + XB > AB
8
POSTULATE 2
The Segment Addition Postulate
Postulate: If B is between A and C, then AB + BC = AC.
Example: If AB = x , BC = 2x and AC = 12, then, find x, AB
and BC.
C
2x
B
A x
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
12
AB + BC = AC
x + 2x = 12
3x = 12
x = 4
x = 4
AB = 4
BC = 8
9
Apply the Segment Addition
Postulate
Maps
The cities shown on the map lie
approximately in a straight line. Use the
given distances to find the distance from
Lubbock, Texas, to St. Louis, Missouri.
SOLUTION
Because Tulsa, Oklahoma, lies between Lubbock and St. Louis, you can
apply the Segment Addition Postulate.
LS = LT + TS = 380 + 360 = 740
ANSWER
The distance from Lubbock to St. Louis is
about 740 miles.
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
11
Compare segments for
congruence
Plot J(– 3, 4), K(2, 4), L(1, 3), and M(1, – 2) in a coordinate plane. Then
determine whether JK and LM are congruent.
SOLUTION
To find the length of a horizontal segment,
find the absolute
value of the difference of the
x-coordinates of the endpoints.
JK = 2 – (– 3) = 5
Use Ruler Postulate.
12
Compare segments for
congruence (continued)
To find the length of a vertical segment, find the absolute value of the
difference of the y-coordinates of the endpoints.
LM = – 2 – 3 = 5
Use Ruler Postulate.
ANSWER
JK and LM have the same length. So, JK ~
= LM.
13
Homework #2
p. 878: 7-12
p. 12: 1-8, 12-23,
32-34, 37-45
14