Beginning Proofs Level 1 In the following series of videos we will
learn how to write simple two-column proofs.
The two-column proof also known as a statement-reason proof, is the main type of proof that we will
use as we continue learning the concepts in geometry.
Traditionally, students do not encounter formal proofs until they take this geometry course.
In reality if you solved an equation in a beginning algebra course and have justified
steps in those solutions then you already have the experience necessary to create and
present mathematical proofs.
A proof is a convincing mathematical argument.
This means that any person, who understands the terminology, accepts the definitions and
premises of the mathematics involved, and thinks in a logically correct fashion could
not deny the validity of the conclusions drawn.
When describing the solution of an equation as an algebraic proof, the premises for the
proof are the given equation, the properties of equality, and the properties of real numbers.
These premises may be used as justifications for each step in the solution of an equation.
The process is analogous to the process used when writing geometric proofs.
An algebraic proof uses algebraic properties of equality.
Recall from your beginning algebra course that the properties of equality include the
addition property of equality, subtraction property of equality, multiplication property
of equality, division property of equality, reflexive property of equality, symmetric
property of equality, transitive property of equality and substitution property of equality.
An algebraic proof also uses the distributive property.
An important part of writing a proof involves the justification of every step.
For example say that we want to solve the following equation and we are asked to prove
it.
We start our two column proof by writing the equation on the left column as follows; we
will write all of our statements on the left column creating a new line for each statement.
On the right column we will write out our reasons or justifications of the statements
from the left column.
Each statement must be numbered and correspond to a reason which has the same number as the
statement in the left column.
Each numbered statement and reason must originate on the same line.
Usually the first statements in a proof will be given to you in this case we are given
this equation so the justification or reason for this statement will be given equation.
Next we proceed in solving the equation so we first apply the distributive property and
distribute 2 into each term; we then write the reason or justification for this step
in this case we write distributive property since we applied the distributive property
to obtain the new statement.
Next we subtract 2 from both sides of the equation and simplify, now this is where many
students make a very common mistake, many students would write subtracted 2 from both
sides as a reason or justification, this would be an incorrect reason, the correct reason
needs to make use of the properties of equality in this case we used the subtraction property
of equality.
The justification for each step is not simply a description of the operation but rather
a general mathematical principle, so make sure you use a mathematical principal as a
justification.
The last step is to divide both sides of the equation by 6 doing that we obtain x equals
5, for our justification we write Division property of equality and not divide by 6.
Like Algebra, geometry also uses numbers, variables and operations.
For example, segment lengths and angle measures are numbers.
So you can use these same properties of equality to write algebraic proofs in geometry.
We will begin to formalize the study of geometry by going over very basic proofs and then add
subsequent layers of complexity as we continue with our studies of geometry.
The geometry concepts presented in these videos are largely based on the ideas set forth more
than 2000 years ago by the Greek mathematician Euclid.
His series of books, Elements, is the first known work in which a logical, deductive system
of reasoning is used as a means of unifying all mathematical knowledge.
In that spirit, geometric proofs will help us learn how to build a logically coherent
system of facts by using deductive reasoning.
Geometry is based on an axiomatic system.
A modern axiomatic system begins with a small set of undefined terms and builds through
the addition of definitions and postulates to the point where many rich mathematical
theorems can be proved.
In an axiomatic system, we agree to leave some terms undefined and then build definitions
of other terms and postulates from those undefined terms.
Undefined terms can be described but cannot be given precise definitions using simple
known terms.
Three undefined terms in geometry are point, line, and plane.
Recall that a point is a circular dot that is shrunk until it has no size.
A line is formed by an infinite number of points, it also has an infinite length so
we denote this idea by using arrowheads; also a line has no thickness.
A plane is a sheet of paper formed by an infinite number of points with no thickness and extending
infinitely in all directions.
Although a plane has no edges, we usually picture a plane by drawing a four-sided figure.
We often label a plane with a capital letter.
These descriptions provide a way for us to visualize points, lines and planes, but they
are not definitions.
We define space to be the set of all points.
Any collection of points is called a geometric figure.
In particular, lines and planes are composed of points, and hence are geometric figures.
Postulates are statements that we assume to be true.
Postulates state relationships among defined and undefined terms.
The purpose of stating postulates is to establish some first principles upon which the subject
of geometry is based.
After introducing postulates and some definitions, we may deduce new results.
Results that we deduce from undefined terms, definitions, postulates or results that follow
from them are commonly called theorems.
A theorem is a mathematical statement that can be proved.
Geometric proofs provide the logical structure that supports an increasingly complex and
powerful set of theorems.
In a general geometric proof, the premises are definitions, postulates, properties and
previously proven theorems.
The early theorems in these videos may seem trivial and writing proofs for such apparently
obvious statements can seem strange at first but it is important to understand that theses
theorems will be needed for later work.
The goal of this and subsequent videos is to provide an opportunity so you can become
familiar with the key elements of a proof.
These proofs will become more complex as we continue with our studies of Geometry.
When writing a geometric proof, you use deductive reasoning to create a chain of logical steps
that move from the hypothesis to the conclusion of the conjecture you are proving.
By proving that the conclusion is true, you have proven that the original conjecture is
true.
When writing a proof, it is important to justify each logical step with a reason.
You can use symbols and abbreviations, but they must be clear enough so that anyone who
reads your proof will understand them.
A geometric proof begins with GIVEN and PROVE statements, which restate the hypothesis and
conclusion of the conjecture respectively.
In a two-column proof, you list the steps of the proof in the left column.
You then write the matching reason for each step in the right column.
Before you start writing a proof, you should plan out your logic.
If a diagram for a proof is not provided, draw your own and mark the given information
on it.
But do not mark the information in the prove statement on the diagram since this is what
we trying to prove.
Alright in our next video we will present and prove our first two theorems in geometry.
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