A transversal intersects two given lines in such a way that the interior angles on the same side of  the transversal are equal. Is it always true that the given lines are parallel? If not, state the condition (s) under which the two lines will be parallel.

Given: AB and CD are two lines intersected by a transversal EF.
   ∠BPQ = ∠PQD.
To prove: AB and CD need not be parallel. When will AB and CD be parallel?
Proof: If the two lines are parallel. Then,
  ∠BPQ + ∠PQD = 180° (Sum of the interior angles is 180°)------------ (1) 
     .^.. If ∠ BPQ = ∠PQD, then (1) need not be true.
.^.. Conditions under which the two lines are parallel is ∠BPQ + ∠PQD = 180°
    Since ∠ BPQ = ∠PQD
                    ∠BPQ = ∠PQD = = 90°
.^.. The two lines should be perpendicular to the transversal.
For the two lines to be parallel, any pair of corresponding angles made by the transversal should be equal, or any pair of adjacent angles on the same side of the transversal will be supplementary.
∴ Two lines that are respectively perpendicular to two parallel lines are parallel to each other.