Geometry: Beginning Proofs (Level 3) | Examples

Beginning Proofs (Level 3) | Examples

In the previous video we proved our first 2 theorems in geometry the right angles theorem and straight angles theorem. We can use previously proven theorems to prove new theorems in geometry. The purpose of a theorem is to shorten your work in a proof. Once you have proven a theorem, you can use it as a reason in later proofs.
Let’s go over the next example to illustrate how this is done.

Here we are provided with a diagram and are given 2 statements about the diagram we are asked to conclude or proof that angle A is congruent to angle C. Notice that in this example we are given the conjecture to prove. We are also provided with the diagram. All we need to do now is to mark the given information in the diagram and plan our argument to prove the conjecture.
So let’s go ahead and mark angle A and angle C as right angles.

Next we will prove the conjecture by using a two column proof. Let’s start with the given information, so we start with angle A is a right angle and the reason is that it is given. Next we write angle C is a right angle and the reason is because it is also given. With these two statements we can now conclude that angel A is congruent to angle C. The reason is because if two angles are right angles, then they are congruent. Notice that this reason is essentially the right angles theorem that we proved in the previous video so we can essentially say right angles theorem as our reason but you should not use this shortcut until you practiced various proofs since taking short cuts will make it harder for you to learn the concepts of geometry.

At times you may need to make assumptions in a diagram and this can be used as a reason as you are proving a conjecture let’s illustrate this with the next example.

In this problem we are provided with a diagram and are asked to prove that 2 particular angles are congruent. Since we are provided with the diagram and no other statement is given we start our proof by writing “diagram as shown” and the reason is because it was given. Next we need to make an assumption about the diagram recall from the previous video series that if 3 points are collinear then we can assume that they form a straight angle. So the next statement we will make is that angle EFG is a straight angle the reason is because it is assumed from the diagram. In the same manner angle HFJ is also a straight angle because it is assumed from the diagram. Finally having established that fact we can conclude that angle EFG is congruent to angle HFJ because if two angles are straight angles, then they are congruent. This is the straight angles theorem that we proved in the previous video.

In some proofs you may need to make use of the properties of real numbers, variables, and operations in your proof let’s illustrate this with the next example.

Here we are given 3 statements along with a diagram we are asked to prove that 2 angles are congruent. As always let’s start by marking the given information on the diagram in this case angle RST measures 50 degrees, angle TSV measures 40 degrees and angle X is a right angle.

Next we proceed with planning the argument. Let’s start by writing down the given information in this case angle RST measures 50 degrees the reason is because it is given, also angle TSV measures 40 degrees and this is also given. Since we are trying to prove that angle RSV is congruent to angle X we need to show that angle RSV has the same measure as angle X. Notice that angle RSV measures 90 degrees notice that if we take the measurement of angle RST and add it to the measurement of angle TSV we get 90 degrees. So for our reason we write addition and write in parenthesis 50 degrees plus 40 degrees equals 90 degrees to show which angles were added together. Now that we have established the fact that angle RSV measures 90 degrees we can write angle RSV is a right angle. The reason is because of the definition of right angles in this case if an angle is a 90 degree angle, it is a right angle.
Now let’s write the statements for angle X, we know that angle X is a right angle and the reason is because it is given. Finally we can conclude that angle RSV is congruent to angle X because of the right angles theorem that says that if two angles are right angles, then they are congruent and this ends the proof.

Alright and this ends the beginning proof series keep in mind that this is only the beginning when it comes to proofs in geometry we will be proving more complicated conjectures as we continue going over the concepts of geometry. In our next series we cover divisions of segments and angles.