Geometry
Lesson 2 – 5
Postulates and Paragraph
Proofs
Objective:
Identify and use basic postulates about points, lines, and planes.
Write paragraph proofs.
Postulate
Postulate (or axiom) – a statement that is
accepted as true without proof.
Postulates
Postulate 2.1

Through any two points, there is exactly
one line.
Postulate 2.2

Through any three noncollinear points,
there is exactly one plane.
Postulate 2.3

A line contains at least two points.
Postulate 2.4

A plane contains at least three noncollinear points.
Postulate 2.5

If two points lie in a plane, then the entire line
containing those points lies in that plane.
Intersection of Lines and Planes
2 lines

If two lines intersect, then their intersection is
exactly one point.
2 planes

If two planes intersect then their intersection is a
line.
Explain how the picture illustrates that each
statement is true. Then state the postulate
that can be used to show each statement is
true.
Line m contains points F and G.
Point E can also be on line m.
Points E, F, and G are all on line m.
Postulate: A line contains at least 2 points.
Lines s and t intersect at
point D.
Lines s and t meet at point D.
Two lines intersect at a point.
Determine whether each statement is
sometimes, always, or never true.
Explain your reasoning.
If two coplanar lines intersect, then the point of
intersection lies in the same plane as the two
lines.
Always, if two points lie in a plane, then the entire
line containing those points lies in that plane.
Four points are noncollinear
Sometimes, a line contains at least 2 points,
but the other points may or may not be on the
same line.
Two lines determine a plane
Always, there are at least 3 noncollinear points
to determine a plane.
Three lines intersect in two points.
Never, cannot meet at 2 points.
If plane T contains EF and EF contains
point G, then plane T contains point G.
Always, if 2 points lie in a plane then the line
Containing those points lies in the plane.
GH contains thee noncollinear points.
Never, noncollinear points cannot be on the same line.
Proof
Proof – used to prove a conjecture
using deductive reasoning to move from
the hypothesis to the the conclusion of
the conjecture you are trying to prove.

Logical argument in which each statement
you make is supported by a statement that
is accepted as true.
General outline
Write a proof
Given that M is the midpoint of XY write a
paragraph proof to show that XM  MY
Step 1: draw a picture to help
Proof:
Given: M is the midpoint of XY.
Prove: XM  MY
If M is the midpoint of segment XY, then by definition
of midpoint XM = MY. By definition of congruence if
the measures are equal, then the segments are
congruent.
Thus, XM  MY
Theorem 2.1
Midpoint Theorem

If M is the midpoint of AB, then AM  MB
Homework
Pg. 128 1 – 13 all, 16 – 30 E