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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.

Congruent triangles ABC and DEF
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.

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