Equal, Congruent, or Similar
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It is important that we understand the difference between these three concepts and their accompanying symbols. Therefore, we'll spend this section discussing the differences.
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Equality
Normally, in mathematics, when we say something equals something else, we mean that they have the same value. As in an equation, if a = b, then the left side (a) has the same value as the right side (b). If I told you that a = b, then if I knew that a = 5 I would automatically know that b = 5.
In Geometry, this isn't quite the case. Geometry is different in that we are talking about shapes, distances, objects, or other such things that are measureable. In which case, saying that one object or shape is equal to another is saying that they are, in fact, the exact same shape, just perhaps moved from one spot to another.
The symbol, that everyone is familiar with, is of course:
Congruence & Similarity
Congruence & Similarity are concepts used in Geometry to describe the size and shape of two different objects and how they are related.
If two different objects are the same size and shape, but are distinctly different shapes, then we say they are congruent. The symbol we use is then:
If two different objects are the same shape but are different sizes, we say that they are similar. The symbol(s) we use to describe similarity are:
Congruency
Congruent shapes are the same size with corresponding lengths and angles equal. In other words, they are exactly the same size and shape. They will fit on top of each other perfectly. Therefore if you know the size and shape of one you know the size and shape of the others. For example:
Each of the above shapes is congruent to each other. The only difference is in their orientation, or the way they are rotated. If you traced them onto paper and cut them out, you could see that they fit over each other exactly.
Having done this, right away we can see that, though the angles correspond in size and position, the sides do not. Therefore it is proved the triangles are not congruent.
Similarity
Similar shapes are like congruent shapes in that they must be the same shape, but they don't have to be the same size. Their corresponding angles are congruent and their corresponding sides are in proportion.
