1
Lesson 1-2
Use Segments
and Congruence
Warm Up Problems
2
Solve for x.
214102 xx
6084163 xx
xx 3101782
3
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
4
Postulate or axiom
Definition: A rule that is accepted without proof.
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.
5
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:
6
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”)
7
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 │
8
Between
Definition: B is between A and C if AB + BC = AC.
AB + BC = AC AX + XB > AB
A BX
9
POSTULATE 2 The Segment Addition Postulate
If B is between A and C, then AB + BC = AC.Postulate:
Example: If AB = x , BC = 2x and AC = 12, then, find x, AB and BC.
AB + BC = AC
x + 2x = 12
3x = 12
x = 4
2xx
12
x = 4AB = 4BC = 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
AC
B
Apply the Segment Addition Postulate
10
SOLUTION
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.
Because Tulsa, Oklahoma, lies between Lubbock and St. Louis, you can apply the Segment Addition Postulate.
LS = LT + TS = 380 + 360 = 740
The distance from Lubbock to St. Louis is about 740 miles.
ANSWER
11
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: )
Compare segments for congruence
12
SOLUTION
To find the length of a horizontal segment, find the absolute value of the difference of the x-coordinates of the endpoints.
Use Ruler Postulate.JK = 2 – (– 3) = 5
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.
Compare segments for congruence (continued)
13
To find the length of a vertical segment, find the absolute value of the difference of the y-coordinates of the endpoints.
Use Ruler Postulate.LM = – 2 – 3 = 5
JK and LM have the same length. So, JK LM.=~
ANSWER
Homework #2
p. 878: 7-12
p. 12: 1-8, 12-23, 32-34, 37-45
14