Question

Why is it important the order we write the vertices of a triangle in a congruent statement?

Answer 1

Because the angles/edges in corresponding positions for each triangle are the ones we are saying are congruent.

Explanation:

Consider the following triangle congruence statement:

`triangle ABC ~= triangle DEF`

This is actually not one, but six—that's right, six—congruence statements. What are the six statements? They are:

`angle A ~= angle D"          "bar(AB)~=bar(DE)`
`angle B ~= angle E"          "bar(AC)~=bar(DF)`
`angle C ~= angle F"          "bar(BC)~=bar(EF)`

Take the first congruence for example: `angle A ~= angle D`.
This is implied because both `A` and `D` are listed first in their triangles (in the statement `triangle ABC ~= triangle DEF`). Likewise, `B` and `E` are in second postion, so that's how we get `angleB ~= angle E`.

A similar reasoning follows for the sides with corresponding vertices.

Now, if the triangle congruence statement was:

`triangle ABC ~= triangle FED`

then

`angle A` would be congruent to `angle F`,
`angle B` would be congruent to `angle E`, and
`angle C` would be congruent to `angle D`.

You can see how naming the triangles in a different order can change which angles/edges are implied to be congruent. This is why we must be careful to name the triangles in corresponding order. Otherwise, we may be saying two angles (or line segments) are congruent, when in fact, they are not.