Explanation: For the explanation follow the detailed proof below:

In order to solve for  and  below, we need to use the fact that similar triangles are proportional.  This allows us to set up ratios of the lengths of the sides to solve for the unknown variables:

To solve for :

So we can set up the following ratios

  by cross-multiplying to get rid of the fractions

So 

To solve for :

So we can set up the following ratios

 by cross-multiplying to get rid of the fractions

So 

Since  is parallel to , the point where  intersects  forms two pairs of opposite vertical angles.  We can now say that  and  are equal by the definition of opposite vertical angles.  Notice that  and  are alternate interior angles.  By definition, .  

We have two equal corresponding angles.  The Angle-Angle (AA) Theorem for similar triangles says that if two triangles have two pairs of congruent corresponding angles, the triangles are similar.  So by AA Theorem, triangles  and  are similar.

Since  is parallel to , the point where  intersects  forms two pairs of opposite vertical angles.  We can now say that  and  are equal by the definition of opposite vertical angles.  Notice that  and  are alternate interior angles.  By definition, .  We have two equal corresponding angles.  The Angle-Angle (AA) Theorem for similar triangles says that if two triangles have two pairs of congruent corresponding angles, the triangles are similar.  So by AA Theorem, triangles  and  are similar. In order to solve for , we need to use the fact that similar triangles are proportional.  This allows us to set up ratios of the lengths of the sides to solve for :

We are able to set up the following ratios:

 cross multiplying to get rid of the fractions

So the length of 

According to the Side-Angle-Side (SAS) Theorem for similar triangles, if two sides of one triangle are proportional to two corresponding sides of another triangle, and the angle between these two sides of known measure is the same for each triangle, then these triangles are similar triangles.

We see that the side of measure 4 and the side of measure 8 are proportional because 8 is simply 4x2.  The same can be seen with the sides of measures 6 and 12, 12 is simply 6x2.  Then the angle between each of these sides on either triangle is the same.  So by SAS Theorem for similar triangles, these two triangles are the same.

Three of these theorems do prove triangles to be congruent; SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and SSS (Side-Side-Side).  The AA (Angle-Angle) Theorem states that two triangles with two congruent, corresponding angles are similar NOT necessarily congruent.

While all of these theorems can prove two triangles to be congruent the Hypotenuse-Leg Theorem (HL) is the only theorem out of these that can only prove two right triangles to be congruent. This theorem states that if two right triangles have one congruent leg and a congruent hypotenuse then they are congruent.

In order to prove anything, we must first state the given information.  This will allow us to move forward with the proof and make different connections based on the given information.

This can be represented by the following equation