Congruency and Similarity (saying that two things are congruent or similar) are two words that mathematicians use to comment on how two shapes are related. They appear a lot in proofs and constructions, and sometimes in everyday life.
Similarity is when we say that two things are the same shape, but not the same size. This would be a lot like saying that, for example, that a boy looks like his father, although he is only one half as tall (though this is not a perfect example). Another example would be saying that a square cardboard box is similar to a smaller square cardboard box.
A major thing to remember about similarity, though, is that the shapes have to be exactly the same! This means that the angles must be the same and in the same order, and that all the sides on the smaller figure must be in the same proportion to the sides on the larger figure. For example, consider the diagram below.
These two squares are similar. All of the angles are the same (90, 90, 90, 90 (the smaller squares mean 90 degrees)) and all of the side ratios are the same- 2/1=2/1=2/1=2/1 (they all equal 2). As a matter of fact, all squares are similar, all circles are similar, all equalteral triangles are similar, and every regular shape of a given type is similar (Regular meaning that all the sides and all the angles are of the same measure, respectively). This is a good thing to know, since it may come in handy to save time on a test or quiz.
Congruency is when two shapes are exactly the same. It is just like similarity, except that the measures of the sides are also the same, as opposed to being simply in the same ratio to each other. Take another look at the first figure again. As you can see, those two figures are not congruent because the measures of the sides are not the same.
These two squares, though, are congurent, because both have the same side lengths.