Proving Parallel Lines

To demonstrate the parallelism of lines, we must first remember how to identify alternate and corresponding angles.
Let's quickly review the steps for identifying corresponding and alternate angles in a very easy way to remember:
We present to you the method we call "The building"
With this method, we will observe the lines that we think are parallel and imagine a building with small apartments - each line is a building in itself.
In each building, there are two floors and two sides - first or second floor and right or left side.
The corresponding angles are those that inhabit the same floor and the same side.
The alternate angles are those that inhabit different floors and opposite sides.

As we can see, angles 1, 2, 5, 6
are located on the first floor
and angles 3, 4, 7, and 8 on the second.
Clearly, each pair of angles is located in its own building.

Therefore, we can state that:
angles 1 and 5, for example, are corresponding angles, they are on the same floor and on the same side,
just like angles 3,7 2,6 and 4,8.
We can also say that angles 1,8 are alternate angles since they appear on different floors and on different sides,
clearly also angles 2,7 3,6 or 4,5.

Great! Now that we remember how to identify corresponding and alternate angles with the building method, we will move on to demonstrate the parallelism of lines.
To prove that the lines are parallel we will see if the following rule is met:

If so, the lines are parallel!
If not, they are not parallel.
Important note: Corresponding or alternate angles are formed only and exclusively by the same transversal.

Observe
Sometimes, the angles will be presented to us through numerical data. To see if the lines are parallel we will simply compare the corresponding or alternate angles and see if they are congruent.
But, in certain cases, there will be unknowns among the angle data.
Remember that adjacent angles equal $180º$, let's complete the missing pieces of the puzzle and compare the expressions with unknowns to find the values at which the lines are parallel.

You can also demonstrate the parallelism of lines through consecutive angles.
Remember: consecutive angles, according to the building method, are those that are on the same side but not on the same level.
If, between two lines and a transversal, supplementary consecutive angles are formed, you can determine that the lines are parallel.

Let's see an example of the demonstration of the parallelism of lines:
We have two lines that we believe to be parallel and a transversal.
The data is presented in the illustration as follows:

The angles indicated in the illustration that measure $100º$ and $80º$ degrees are neither alternate nor corresponding.
Therefore, we will try to complete the puzzle pieces and see where we get.
We know that adjacent angles equal $180º$ degrees and, consequently, we can complete the scheme as follows:

Great! We have even obtained two pairs of congruent alternate angles, so, we can determine that the lines are parallel. (One pair would have been enough).

Observe :
If you had wanted to use the demonstration of consecutive angles you could have done it very easily.
We see in the first illustration, that the given angles are consecutive angles (they are on the same side but not on the same level).
We can easily see that together they measure $180º$ and, consequently, we can determine that the lines are parallel.

If you are interested in this article, you might also be interested in the following articles:

Parallel Lines

Parallelogram - Checking the Parallelogram

The area of the parallelogram: what is it and how is it calculated?

Ways to identify parallelograms

Rotational symmetry in parallelograms

In the blog of Tutorela you will find a variety of articles about mathematics.