Find the value of x, given that lines a and b are parallel, and lines c and d are parallel.

Based on the image, we have two parallel lines ( a ) and ( b ), intersected by two other parallel lines ( c ) and ( d ). We are given that one of the angles is ( 105^{\circ} ) and we need to find the value of the angle ( x ).

When parallel lines are intersected by another line, corresponding angles are equal. Also, supplementary angles add up to ( 180^{\circ} ).

Since line ( c ) intersects parallel lines ( a ) and ( b ), the angle corresponding to the ( 105^{\circ} ) angle is also ( 105^{\circ} ) on line ( a ).

Now, line ( d ) intersects parallel lines ( a ) and ( b ). The angle ( x ) and the ( 105^{\circ} ) angle are supplementary angles because they lie on the same side of the transversal line ( d ) and between the parallel lines ( a ) and ( b ). Therefore, their sum is ( 180^{\circ} ).

So, we can write the equation:

Solving for ( x ):