Beginning Proofs.
Level 2.
In the previous video we introduced the basic elements of a geometric proof.
In this video we will present and prove our first two theorems in geometry.
Keep in mind that your class or textbook might use a different numbering system than the
one used in these videos when referring to theorems in geometry; whenever possible certain
theorems will have a name associated with them.
Ok let’s take a look and prove our first theorem.
Theorem 1 Right Angles Theorem.
If two angles are right angles, then they are congruent.
We start the two column proof by writing down the conjecture to be proven in this case if
two angles are right angles then they are congruent.
Next we draw a diagram to illustrate the hypothesis; the hypothesis is the statement that begins
with the word “if” in this case “if two angles are right angles”.
This statement also represents the given information of our proof so let’s draw two angles and
call them angle A and angle B. Let’s also mark them as right angles since this is also
given information.
Next we state the conclusion of the conjecture, the conclusion will usually be the statement
followed by the word “then” in this case it is “then they are congruent”.
We are essentially trying to prove this statement.
Using our diagram we translate this statement by writing angle A is congruent to angle B.
Now that we have the diagram, the given information and what we are trying to prove we can now
proceed with the planning part to prove this theorem.
Similar to an algebraic proof, we usually start with the given information, in this
case we are given that angle A is a right angle.
The reason for this first statement is simply that it is given.
Next we write that the measure of angle A is equal to 90 degrees.
The reason for this statement is because of the definition of right angles in this case
if an angle is a right angle, then its measure is 90 degree.
Next we write angle B is a right angle and the reason for this statement is that it is
also given.
We then write the measure of angle B is equal to 90 degrees.
The reason for this statement is once again because of the definition of right angles
so we can either write the reason from line 2 or just write same as line 2.
The final statement would be angle A is congruent to angle B. The reason for this statement
is because of the definition of congruent angles, if two angles have the same measure,
then they are congruent.
Notice that we established the fact that angle A and angle B both measure 90 degrees in line
2 and line 4, with this last reason we officially end the proof.
The proof process can be broken down into 5 steps.
In the first step we write the conjecture to be proven this step is usually written
for you in the form of a theorem that you need to prove but in the event that it is
not provided to you, you will need to write it down.
Step 2: if a diagram is not provided for you then you need to represent the hypothesis
of the conjecture by drawing a diagram, remember the hypothesis is usually the statement after
the word “if”.
Step 3: state the given information and mark it on the diagram.
For example if an angle or side is congruent to another angle or side then mark them on
your diagram for easy reference.
Step 4: state the conclusion of the conjecture in terms of the diagram, remember this statement
is usually the statement after the word “then”.
Steps 1 through 4 are usually provided to you in most homework problems and test, in
this course we will mainly focus on the final step.
Step 5: plan your argument and prove the conjecture.
This is by far the hardest step for many students in an introductory geometry course, as with
all math problems practicing how to prove conjectures is essential in becoming better
at them.
Alright let’s take a look at the next theorem and prove it.
Theorem 2 Straight Angles Theorem.
If two angles are straight angles, then they are congruent.
Let’s start by drawing two straight angles let’s label them as angle ABC and angle
DEF.
This will also be our given information.
Our conclusion or what we are trying to prove is that angle ABC is congruent to angle DEF.
Alright we will start the proof by writing down the given information in this case angle
ABC is a straight angle and the reason is that it is given.
Also the measure of angle ABC is 180 degrees the reason is because of the definition of
a straight angle in this case if an angle is a straight angle, then its measure is 180
degrees.
Next we write angle DEF is a straight angle and the reason is that it is given.
Also the measure of angle DEF is 180 degrees and the reason is the same as line 2 because
of the definition of straight angles.
Finally we can conclude that angle ABC is congruent to angle DEF because of the definition
of congruent angles.
In this case if two angles have the same measure, then they are congruent.
We established the fact that angle ABC and angle DEF both measure 180 degrees in line
2 and line 4 and this ends the proof.
Alright and these are the first two theorems that we will be using to prove other theorems
later on in this course in our next video we will go over a couple of examples illustrating
how to tackle more challenging two column proofs.