Question - Determining the Angle Value in Parallel Lines with Transversal

Given that \( AB \parallel CD \) and the angles provided in the figure, we can use the properties of alternate interior angles and corresponding angles to solve for \( x \).

Since \( AB \parallel CD \), angles \( \angle ABC \) (75 degrees) and \( \angle BCD \) are corresponding angles and thus they are equal.

\( \angle ABC = \angle BCD = 75^\circ \)

Similarly, \( \angle BCD \) and \( \angle CDE \) are alternate interior angles with the line \( DE \) being a transversal intersecting the parallel lines \( AB \) and \( CD \). Therefore, they are also equal.

\( \angle BCD = \angle CDE = 75^\circ \)

Using the fact that the sum of angles in a triangle is \( 180^\circ \), we have the following equation for triangle \( CDE \):

\( \angle CDE + \angle DCE + \angle CED = 180^\circ \)

Substitute the known angle values:

\( 75^\circ + 30^\circ + x = 180^\circ \)

Solve for \( x \):

\( x = 180^\circ - 75^\circ - 30^\circ \)

\( x = 75^\circ \)