Why are opposite triangles in a parallelogram congruent?

Question

If you draw a parallelogram its diagonals will form four triangles of which at least the opposite pairs will be congruent, which is to say, each triangle is a reflection of its opposite one, why is this? What property of parallelograms am I missing here? because if I draw a trapezium its diagonals will obviously not form congruent triangles.

Answer 1

1. Opposite sides of a parallelogram are congruent.
2. The diagonals bisect each other.

Using these two properties alone, you know that by $SSS$, the opposite triangles which are formed by the diagonals of the parallelogram are congruent.

The following may not apply for a trapezoid because of the fact that the opposite sides of a trapezoid may not be congruent, so you cannot prove the same by $SSS$, as one side is not congruent to the other.

Answer 2

Another way to see this is to use affine geometry properties. Any parallelogram is affinely equivalent to a square, and the opposite triangles property holds for any square. Note that if two line segments on the same or parallel lines are congruent, then they remain congruent after any affine transformation. Also note that a trapezium that is not also a parallelogram is not equivalent to a prallelogram.