Segments AB and CD intersect at point M, which is the middle of each of them. Prove that AC is parallel to BD.

1. Consider △ AMC and △ BMD.

In them AM = BM and CM = DM by condition. ∠AMC = ∠BMD as they are vertical.

Hence, △ AMC = △ BMD on two sides and an angle between them.

2. Let us prove that AC ∥ BD.

The equality of the triangles △ AMC and △ BMD implies the equality of their corresponding elements. Hence, ∠CAM = ∠DBM.

∠CAM and ∠DBM are crossed for lines AC and BD and secant AB. Since the angles lying crosswise are equal, the lines AC and BD are parallel.