Geometry: CPOCTAC

CPOCTAC

Two triangles are congruent if one fits perfectly over the other. It should come as no surprise that you have to be specific about exactly how these two triangles fit together. After all, there are three angles, and three sides, and you wouldn't want to pair wrong angles or the wrong sides together. Figure 12.1 shows two triangles: ΔABC and ΔDEF. These two triangles look like identical twins. You can say that ΔABC and ΔDEF are congruent (and write ΔABC ~= ΔDEF ) if ∠A ~= ∠D, ∠B ~= ∠E, ∠C ~= ∠F, and ¯AB ~= ¯DE , ¯AC ~= ¯DF, and ¯BC ~= ¯EF. In this situation, vertex A corresponds to vertex D, vertex B corresponds to vertex E, and vertex C corresponds to vertex F.

Figure 12.1ΔABC and ΔDEF are congruent triangles.

The order that you write the vertices of the triangle shows which angles and which sides are congruent. You can also talk about corresponding angles: ∠A and ∠D are corresponding angles, as are ∠B and ∠E, and ∠C and ∠F. This definition of congruence works both ways. If you are told that ΔABC ~= ΔMNO, then right away you know that their corresponding parts are congruent: for example, ∠A ~= ∠M and so on. The notion that corresponding parts of congruent triangles are congruent will be used so often that it will be abbreviated CPOCTAC. It will play a crucial role in proving the congruence of line segments and angles.

As you have already seen with congruent line segments and congruent angles, the notion of congruence with respect to triangles is an equivalence relation. It has the reflexive, symmetric, and transitive properties, which are easy to verify.