What is the difference between equality and congruency? When should I say that two figures are congruent and when that they are equal?
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1$\begingroup$ A bagel might be congruent to a donut, but they certainly won't be equal. $\endgroup$– Gerry MyersonCommented Sep 1, 2014 at 13:18
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1$\begingroup$ 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. $\endgroup$– Andreas BlassCommented Dec 9, 2016 at 2:22
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2 Answers
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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.
answered Sep 1, 2014 at 13:16
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2$\begingroup$ 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. $\endgroup$ Commented Sep 1, 2014 at 13:21
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$\begingroup$ @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. $\endgroup$ Commented Sep 1, 2014 at 13:28
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$\begingroup$ 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.) $\endgroup$ Commented Sep 1, 2014 at 13:40
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$\begingroup$ What about if we have an angle ABC and an angle ABD, where D is a point on the line BC. Are these angles equal? $\endgroup$ Commented Sep 14, 2020 at 3:21
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$\begingroup$ @AaronFranke: Under standard definitions of "angle" (such as Wikipedia's in their "Angle" article), an angle is defined by two rays with a common endpoint. The angle determined by three points is actually defined by the rays defined by those points. So if your point A is distinct from points B, C, and D, then yes, angle ABC (or more precisely, the angle defined by rays AB and AC) equals angle ABD (again, the angle defined by rays AB and AD). $\endgroup$ Commented Oct 23, 2020 at 23:37
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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.
