Question

given ( d||e ), what is ( mangle x )?
(the diagram is not drawn to scale.)
( 60^{circ} )
( 70^{circ} )
( 50^{circ} )
( 80^{circ} )

Explanation:

Step1: Find the supplementary angle

The angle adjacent to \(110^{\circ}\) is \(180 - 110=70^{\circ}\) (supplementary angles sum to \(180^{\circ}\)).

Step2: Use the triangle - like angle relationship

Since \(d\parallel e\), by the properties of parallel lines and transversals, we know that \(x + 50=70 + 40\) (the sum of angles in a "virtual" triangle - like structure formed by the transversals). Wait, another way: Using the exterior angle property (or angle - chasing with parallel lines). The angle formed by the transversal with \(e\) (the non - \(110^{\circ}\) adjacent angle is \(70^{\circ}\)). Then, considering the angles around the intersection on line \(d\), we have \(x+50 = 70\) (alternate interior angles and angle addition). So \(x=70 - 50\).
\(x = 60^{\circ}\) (using the property that the sum of angles in the "Z - like" or transversal - related angle chasing. Let's re - explain:
Let \(y\) be the angle adjacent to \(110^{\circ}\) on line \(e\), \(y = 180-110 = 70^{\circ}\).
We know that \(x+50=y\) (by the property of parallel lines \(d\parallel e\) and the transversal, the sum of \(x\) and \(50^{\circ}\) is equal to the angle \(y\) which is \(70^{\circ}\) when considering the "corresponding - like" or "alternate - interior - like" angle relationships in the transversal - cut parallel lines structure). So \(x=70 - 50\).

Answer:

\(60^{\circ}\) (Option A)