Segments МР and ЕК intersect in their middle O. Prove that ME is parallel to PK.

Triangles MOE and РOC are equal according to the first sign of 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.

MO = OP, EO = OK – by condition;

angle MOE = angle РOK – as vertical (vertical angles are equal).

From the equality of the triangles MOE and РOK it follows that the angles E and K are equal.

The angles E and K are internal crosswise at the straight lines ME, РK and secant EK.

By the criterion of parallelism of straight lines (If at the intersection of two straight secant lines, the lying angles are equal, then the straight lines are parallel.), Our straight lines ME and РK are parallel, which was required to prove.