### Chapter_2.5_Postulates_and_Paragraph_Proofs

```Chapter 2.5
Postulates and Paragraph Proofs
Objective: Introduce Proofs
Check.1.14
CLE 3108.4.3
Spi.1.4
Identify and explain the necessity of postulates, theorems, and
corollaries in a mathematical system.
Develop an understanding of the tools of logic and proof, including
aspects of formal logic as well as construction of proofs.
Use definitions, basic postulates, and theorems about points, lines,
angles, and planes to write/complete proofs and/or to solve problems.
Postulates
(points, lines and planes)
Copy the following
on a clean sheet of
paper
2.1 Through any two points, there is exactly one line.
2.2 Through any three points, not on the same line, there is exactly
one plane.
2.3 A line contains at least two points
2.4 A plane contains at least three points, not on the same line
2.5 If two points lie in a plane, then the entire line containing those
points lies in that plane.
2.6 If two lines intersect, then their intersection is exactly one point.
2.7 If two planes intersect, then their intersection is a line.
Glossary Section
Postulate (Axiom)
Statement that describes basic relationship of
Geometry
Postulates and Proofs

Determine if the following is always, sometimes
or never true.

If points, A, B, and C lie in a plane M, then they are
collinear. Sometimes

There is exactly one plane that contains noncollinear
points P, Q and R. Always, Postulate 2.2

There are at least two lines through points M and N.
Never, Postulate 2.1
Use the postulates

Determine if the following is always, sometimes
or never true.

If a plane T contains EF and EF contains point G, then
plane T contains point G. Always, Postulate 2.5

For XY, if X lines in plane Q and Y lies in plane R, then
plane Q intersects plane R. Sometimes, if R and Q are
parallel they would not
intersect

GH contains three noncollinear points.
Never, by definition of non-collinear
Proofs

A logical argument in which each statement is
supported by a statement accepted as true.

Paragraph Proof or Informal Proof
1.
2.
3.
4.
5.
Include the theorem or conjecture to be proven
List the given information
Draw a diagram (if possible)
State what is to be proven
Develop a system of deductive reasoning
Proof




1.
2.
3.
4.
5.
Conjecture to prove
Given information
Diagram
Statement
Deductive reasoning
Given that M is the midpoint of PQ, write a
paragraph proof to show that PM  MQ.
M
Q
P
Prove PM  MQ.
Given: M is the midpoint of PQ
From the definition of a midpoint, PM = MQ.
This means that PM and MQ have the same
measure. By the definition of congruence, if two
segments have the same measure then they are
congruent, thus PM  MQ.
Proof

1.
2.
3.
4.
5.
Conjecture to prove
Given information
Diagram
Statement
Deductive reasoning
Given AC intersecting CD, show that A, C, and D
determine a plane.
D
C



Prove ACD is a plane.
Given: AC intersects CD
A
AC and CD must intersect a C because 2.6 if two lines
intersect then their intersection is exactly one point. Point
A is on AC and D is on CD. Therefore A and D are not
collinear. Therefore ACD is a plane as 2.2 states that any
three noncollinear points define a line.
Proof:
?
A.
B.
C.
D.
Definition of midpoint