Prove that if at the intersection of 2 intersecting lines, the lying angles are equal, then the lines are parallel.

Let lines a and b be parallel, c – secant.
Line a intersects with line c at point C1, with line b at point C2.
Through point M – the middle of the segment C1C2, draw a perpendicular to the line a AB.
Consider triangles AC1M and BC2M.
These triangles are equal in side and two adjacent corners, therefore, in these triangles the corresponding elements are equal. Hence, <C1AM = <C2BM = 90 °
Consequently, the straight line AB, intersecting the straight line a at a right angle (by construction) will also intersect the straight line b at a right angle (from what was proved). Therefore, straight lines a and b are parallel.