Geometry: Beginning Proofs (Level 2) | Right and straight angles theorem

Beginning Proofs (Level 2) | Right and straight angles theorem

In the previous post we introduced the basic elements of a geometric proof. In this post 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 post 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.

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 then 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.

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 post we will go over a couple of examples illustrating how to tackle more challenging two column proofs.