The segments AB and CD intersect in their common middle. Prove that lines AC and BD are parallel.

Having made the diagram, you can see the formed triangles CAO and BDO. (O is the intersection point). Consider them:
1) Angle BOD = COA (property of vertical angles);
2) CO = OD (O – middle);
3) AO = OB (O – middle).
Hence, triangles CAO and BDO are equal. When added, they will match. Therefore AC = BD and they are equally distant from O, so they are parallel.