# What is the difference between congruency and equality?

What is the difference between equality and congruency? When should I say that two figures are congruent and when that they are equal?

• A bagel might be congruent to a donut, but they certainly won't be equal. – Gerry Myerson Sep 1 '14 at 13:18
• There's a good example already in the title of your question: The three e's in "between" are congruent (unless your browser does something very strange) but they are not equal. If they were equal, there would be only one of them. – Andreas Blass Dec 9 '16 at 2:22

## 4 Answers

In geometry, a "figure" is a set of points in the plane. So, two figures are equal if they have the same points. In other words, two equal figures are exactly equal: the same figure.

Congruent figures have the same shape and size (informally) but possibly different points.

No diagram is needed for this explanation.

Part of the confusion between "equal" and "congruent" probably comes from the fact that congruence is an equivalence relation. In other words, congruence satisfies many of the properties of equality. So if you think of congruent figures as equal, your reasoning may well not lead you astray. However, equivalence is not quite the same as equality, so sloppiness here may have consequences.

• To avoid ambiguity re whether similar figures are the same shape, one might want to emphasize same shape and same size, in Euclidean geometry anyway. – Travis Sep 1 '14 at 13:21
• @Travis: Yes, you are correct, I messed up. I'll change my answer. I did try to stick to an informal level, in keeping with the question. – Rory Daulton Sep 1 '14 at 13:28
• I didn't read it as a mistake, just as potentially ambiguous for someone who's not familiar with the terms. (Anyway, in some sense it's arbitrary that Euclidean geometry is the default for this, rather than, say, similarity or affine geometry.) – Travis Sep 1 '14 at 13:40

In everyday language (outside of mathematics/geometry) we almost never use "equal" to mean that something is equal to itself. We use it to talk about two or more separate things, never just one. I was taught to say two line segments or two angles are equal if they have the same measure. In the "olden days" when I was coming along, only closed figures could be said to be congruent. Apparently many textbooks still read this way, for I tutor online and see many cases where students today uses "equal" and "congruent" in the "old" sense. I'd say anybody who gets confused between the two systems isn't really using their noggin.

two figures are congruent if their corresponding parts are of the same measurement. Ex. If two segments have equal length, then they are congruent. It is informal to say that two figures are equal. Two figures are not equal, they are congruent if the coreesponding measurements are equal.

Thomas Jefferson was wrong. All men are not created equal. They are created congruent. They may have equal rights, but no man is equal to another man.