When to use the equal or the congruent sign?

Question

Lets say you're naming segments that are the same, would you write that their equal or congruent?

Lets say you're naming angles that are the same, would you write that their equal or congruent?

Lets say you're naming lines that are the same, would you write that their equal or congruent?

is there any way of knowing which one it is?

Thanks :)

Answer 1

That depends on what do you mean by "the same?" If two segments are actually the same segment, I write that they are equal. If the segments are merely the same _length_, and not the same segment (e.g. they have different endpoints), then they are congruent. Ditto for angles -- if they are actually the same angle, then they are equal. If they are distinct angles that have the same measure, they are only congruent.

The distinction between the two is that to say that two things are equal implies that they are the same thing. Not that they are the same length and measure, or even that they are identical in every way, but that they are THE SAME THING. The notion of congruent refers to objects that have the same size and shape, but may nonetheless be distinct objects (say, because one is over here and the other is over there). This is a weaker notion than equality (although of course, since any object is congruent to itself, equality implies congruence), but more generally useful in geometry (because we need a way to express the idea that two objects have the same size and shape even if they are different objects).

One commonly repeated (even by teachers, who should know better) notion that I would like to point out as false is the idea that the difference between congruence is a special name given to equality of geometric figures. This is wrong -- most of the time, when two objects are congruent, they are nonetheless not the same object and thus not equal. This is not just semantic quibbling either - equality allows some logical operations that congruence does not. For instance, if two objects are equal, one may be substituted for the other in any expression whatsoever without changing the truth value of the expression. This is not true for congruence, as seen in the following figure:

A..... ......C
|....... ......|
|....E ___|___F
|....... ......|
B .......... D

In this figure, AB≅CD, and CD intersects EF. Now, if AB=CD, then we could replace CD with AB in the expression "CD intersects EF" and obtain another true statement. However, "AB intersects EF" is clearly not a true statement, demonstrating that AB≠CD (although you can see that from the diagram).

Conversely, consider the following diagram:

........ .........D
........ ......../\
........ ......./...\
........ ...C/ .....\
....... ...../ .........\
A____/ ............\______F
.......... B ............E

Here, C is supposed to be the point on BD two lines above B. Now, in this case, BD≠BC (indeed, BD≇BC). However, the ray from B through C is the same as the ray from B through D (since a ray, by definition, extends indefinitely in one direction), so we have that ∠ABC and ∠ABD are the same angle. Thus ∠ABC=∠ABD, and thus ∠ABC≅∠ABD as well. We can contrast that with the situation of ∠ABD and ∠DEF -- assuming I've drawn this correctly, they both have the same measure, so ∠ABD≅∠DEF. However, they are not the same angle, (in particular, ∠ABD contains the point C, and ∠DEF does not), so ∠ABD≠∠DEF.

Now that you know the difference between equality and congruence, it should be easy to deduce which one you should write. If you mean to assert that two geometric figures are the same object, write equals. If you mean only to assert that they have the same size and shape, write congruent.

Just two more comments to make before I finish off my answer -- the first is that when naming triangles, the order of the points actually matters (this is so that corresponding sides can be correctly matched when proving congruence). Thus ABC≠CBA, and typically ABC≇CBA either (if ABC≅CBA, this means that AB≅CB, so triangle ABC is isosceles). So be careful about asserting congruences between triangles.

The second point is that "their" is the possessive form of they, and the correct contraction of "they are" is "they're." This is very helpful to know, especially if your English teacher is a pain in the ASCII (I know mine was).

- Pascal

Answer 2

Congruent Sign

Answer 3

You use congruent when saying two shapes are the same; you use equal when saying two numbers or measurements are the same So segment AB is congruent to segment CD; length of segment AB equals length of segment CD

Answer 4

Segments are congruent. Their lengths are equal.

Angles are congruent. Their measures are equal.

If two lines are the same, they are equal.

Congruent deals with geometric figures. Equal deals with numerical values obtained by measurement.