Geometry CPE
Date ___________________
Notes Section 1-4: Measuring Segments
1. A statement accepted true without proof is called a postulate.
2. A statement that we prove to be true is called a theorem.
3. Distance Postulate: To any two distinct points, there exists a real unique real number called the
distance.
a. Distance is always non-negative (unlike displacement in physics, which can be negative).
b. The distance from point A to point B is denoted by AB. The length, or measure of a line
segment, ̅̅̅̅, for example, is also AB.
c. Note that we never discuss the length of a line because its length is infinite.
d. If AB = 0, then A and B must be the same point. This is known as a zero segment.
e. Question: Is AB the same number in a spherical geometry system as it is in a plane system?
4. Ruler Postulate: The points on a line can be put in a one-to-one correspondence with the real
numbers. The real number corresponding to a point is called its coordinate. The distance between
points A and B, written AB, is the absolute value of the difference of the coordinates of A and B.

Every point has a number

Every number has a point

This implies a line is infinitely dense, that is there is no way to pass from one
half plane created by a given line to the other half plane created by that line
without passing through a point of the line
a. Ex: If the coordinate of A is a and the coordinate of B is b, then AB = |a – b| or |b – a|
b. Ex: If the coordinate of A is a and the coordinate of B is 7, then AB = |a – 7|
c. Ex: If the coordinate of A is a and the coordinate of B is 7, and AB = 10, find the value(s) of a:
5. Def: If A, B, and C are collinear in that order, then B is between A and C.
a. Note the difference between your everyday use of the term “between” and the geometric
definition.
C
D
D
b. Is D between A and B?
A
B
C
6. Segment Addition Postulate: If B is between A and C, then…
B
a. AC = AB + BC
b. AB = AC – BC
A
c. BC = AC – AB
7. Def: Segments are congruent if they have the same length: AB = CD ↔ ̅̅̅̅
̅̅̅̅
a. When writing up proofs later in the course, we will often go from “ = ” to “
” or vice/versa –
we will use “definition of congruent segments” as a reason.
b. It is incorrect to write that ̅̅̅̅
̅̅̅̅ or that
; Equals is for numbers, congruence is for
shapes
c. Congruent figures are the same size and shape, but should not be confused with equal. If two
rectangles have equal lengths and equal widths, then the rectangles are congruent, but not
equal. You are congruent to your clone, but you are not equal!
8. Pythagorean Theorem: If a and b represent the lengths of the legs of a right triangle and c represents
the length of the hypotenuse of a right triangle, then a2 + b2 = c2.
9. Distance Formula: On the coordinate plane, if the coordinate of A is (
then the length of ̅̅̅̅, or the distance from A to B, is given by
√(
) and the coordinate of B is (
)
(
a. If the coordinate of A is (6, –8) and B is (–3 , –13), determine the length of ̅̅̅̅:
b. If the coordinate of A is (5, –2) and B is (–2 , r), AB = 25, find the value of r:
)
),