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To prove that two triangles have the same shape, certain parts of one triangle
must coincide with certain parts of the other triangle. Specifically, the
vertices of each triangle must have a one-to-one correspondence. This
phrase means that the measure of each side and angle of each triangle
corresponds to a side or angle of the other triangle. As we will see, triangles
don't necessarily have to be congruent to have a one-to-one
correspondence;
but when they are congruent, it is necessary to know the correspondence of the
triangles to know exactly which sides and which angles are congruent.

As you know, when a triangle's name is derived from the letters given to either
its angles or sides (ex. triangle ABC). Until now, it didn't seem to matter
which letters were there--as long as all three vertices were in the name, we
knew which triangle we were talking about. Now, when we want to say that a
given triangle, like triangle ABC, is congruent to another triangle, like
triangle DEF, the order of the vertices in the name makes a big difference.

When two triangle are written this way, ABC and DEF, it means that vertex A
corresponds with vertex D, vertex B with vertex E, and so on. This means that
side CA, for example, corresponds to side FD; it also means that angle BC, that
angle included in sides B and C, corresponds to angle EF. These relationships
aren't especially important when triangles aren't congruent or similar. But
when they are congruent, the one-to-one correspondence of triangles determines
which angles and sides are congruent.

When a triangle is said to be congruent to another triangle, it means that the
corresponding parts of each triangle are congruent. By proving the congruence of
triangles, we can show that polygons are congruent, and eventually make
conclusions about the real world.