The triangle ABC is given. Is there a triangle ABD, which is equal to triangle ABC? Explain your answer.
There is no such triangle in the plane to which the ABC triangle belongs. If the triangles ABC and ABD exist, the points C and D must be located in the same half-plane about the line AB. According to the seventh Euclid's axiom from any given half-straight line to the given half-plane can be build a given degree angle measure less than 180o, and only one. According to the sixth Euclid's axiom on every half-straight line from its initial point can be build a segment of given length, and only one. Thus we can build one angle equal to ABC only and build one segment equal to BC only. But the ABD triangle, equal to ABC triangle can be build in any other plane.