# Formulate and prove the first criterion for the equality of triangles.

If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles are equal.

Proof:

1) Set aside the angle В1А1С1 in the same plane with the AC boundary, where the angle BAC lies in such a way that the A1С1 side coincides with the AC side;

2) Since the angle BAC = angle B1A1C1, then according to the axiom of laying the angle in a half-plane, the rays AB and A1B1 coincide;

3) According to the axiom of uniqueness of postponing a segment on a ray, point B1 coincides with point B, point C coincides with C1. Therefore, we get that BC = B1C1, and AB = A1B1 by the condition of the theorem, then the angles and sides of triangles ABC and A1B1C1 coincide, so these triangles are equal.