To be congruent, triangles must have all corresponding angles and sides be of the same measure.  Corresponding sides of triangles are sides of the same measure in the same positions on different triangles.  This is shown in red on the two triangles below.  Corresponding angles of triangles are angles of the same measure in the same position on different triangles.  This is shown in blue on the two triangles below.

Shapes, in general, are similar when they are the same shape but have different sizes.  If they were the exact same size, they would be congruent.  To have the same shape, even when the sizes differ, the shape needs to have the same angles.  Take the two triangles below for example, they have equal angles but are different sizes.  Therefore these triangles are similar.

If two right triangles have a congruent leg and hypotenuse we can say they have two congruent sides.  Since both of these triangles are right triangles, we also know that they have a congruent angle (their 90-degree angle).  So these two triangles have two pairs of corresponding congruent sides and one pair of corresponding congruent angles.  By SAS Theorem, these right triangles are congruent.  When using this fact that two right triangles are congruent when they have a congruent hypotenuse and a congruent leg, this is called the HL Theorem.

The Side-Side-Side (SSS) Theorem states that two triangles are congruent if the three corresponding sides of each triangle are congruent.  If you think about it, in order for all three corresponding sides to be equal they must fit together at the same angles as well making these triangles congruent.  A detailed proof based on Philoh’s approach is below for a better explanation.  Note that there are stronger proofs than this for this theorem, but this proof is best visually for this level of understanding of the theorem.

Proof: 

(We want to show that if all three sides of two triangles are equal, then these two triangles are congruent)

Consider two triangles  and .  Assume the following:

Since  we are able to rotate triangle  to coincide  and .  We will just call this common line segment  for simplification.

Now we will draw a line from vertex  to vertex .  This gives us two isosceles triangles.  Recall from the definition of an isosceles triangle that the two adjacent sides and angles are equivalent.  So  and .

We can then deduce that . We now know that at least two sides of the two given triangles are equal and the angle enclosed by these sides is congruent between the two triangles. By SAS Theorem, these two triangles are congruent.

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.