Why does geometry use the congruence symbol rather than the equal sign?

Question

Also why can the measure of 2 angles be equal but the angles are congruent? For example, the measure of angle A is equal to the measure of angle B but angle A is congruent to angle B.

Answer 1

Equal can only refer to numbers - only numbers can truly be equal. Segments might have equal lengths or angles might have equal measures but those are numbers so the segments with equal lengths or angles with equal measures are said to be congruent

Answer 2

It's not such a big deal--it's just that geometric figures are not strictly regular "numbers," and two figures can be identical without being right on top of one another. The term might come from the use of "congruent" in number theory. Two numbers are "congruent, modulo 5" if they have the same remainder when divided by 5. So as far as mod 5 "clock" counting is concerned, 2 and 7 are the same, but they are not in some other more usual sense.

Euclid routinely uses "equal" to describe what you'd call "congruent" figures. But then he also uses "equal" to describe two dissimilar triangles which have the same area. So it's a matter of refining the terminology.

Answer 3

In maths, everything has proper definitions. However, at lower levels, many of these definitions were not taught to avoid making the subjects too boring. Once you know the way things in maths are defined, you will recognize many of these seemingly strange phenomena.

The symbol '=' is used to represent the following relationships that two objects are equals, i.e. exactly the same, where objects included in the definition are complex numbers (thus inclusive of real numbers, integers, etc.), and items described by them, such as matrix, vectors, coordinates, functions, etc. All these objects are considered abstract.

In geometry, we are dealing with actual objects such as angles, shapes, etc. To say two actual objects are equal means they are the same. However, in the real world, there aren't any 2 separate objects being exactly the same, down to the atomic levels. Thus, another term 'congruent' is used to suggest that two objects are the same in the sense that the corresponding measurements used to define them are equal.

Here, I guess angle A and angle B are angles at different locations A and B, respectively.

Answer 4

= means identity (same thing). So 7 = 7, 'cause it's the same thing. Congruence means having the same measurements. So all right angles are congruent, even if one's in Paris and the other is in Hong Kong. To say Angle A and Angle B are equal is to say they're the same angle, not that they have the same measure. It's actually a pretty fine point. Not sure that many besides math teachers care that much. In actual real world situations, it's usually apparent by the context of the situation what you mean.

Answer 5

Your wrong.

Angles are equal. Shapes are congruent.

If for example you have two triangles that have all angles 'equal'. Then you say that the triangles are congruent.

Answer 6

Because congruence is a type of equivalence. It is not complete equivalence. We can say 1=1 because it is a measure of value, but you cannot say a triangle is equal to another triangle because although they may have all the same measurements they are not equal because they are two seperate objects.