Problem #DES-270124

Problem

In a lot of geometric problems the main idea is to find congruent figures. We call two polygons congruent if all their corresponding sides and angles are equal. Triangles are the easiest sort of polygons to deal with. Assume we are given two triangles $ABC$ and $A_1{B}_1{C}_1$ and we need to check whether they are congruent or not, some rules that help are:

• If all three corresponding sides of the triangles are equal, then the triangles are congruent.

• If, in the given triangles $ABC$ and $A_1{B}_1{C}_1$, two corresponding sides $AB=A_1{B}_1$, $AC=A_1{C}_1$ and the angles between them $\mathrm{\angle }BAC=\mathrm{\angle }{B}_1{A}_1{C}_1$ are equal, then the triangles are congruent.

• If the sides $AB=A_1{B}_1$ and pairs of the corresponding angles next to them $\mathrm{\angle }CAB=\mathrm{\angle }{C}_1{A}_1{B}_1$ and $\mathrm{\angle }CBA=\mathrm{\angle }{C}_1{B}_1{A}_1$ are equal, then the triangles are congruent.

At a previous geometry lesson we have derived these rules from the axioms of Euclidean geometry, so now we can just use them.