Question

what is the value of x in the figure shown below? a) 31 b) 38 c) 111 d) 142

Explanation:

Step1: Recall property of parallelogram

In parallelogram \(ABDC\), \(AB\parallel CD\). So, \(\angle BAC=\angle ACD\) (alternate - interior angles).

Step2: Use angle - sum property of triangle

In \(\triangle ABC\), we know that the sum of interior angles of a triangle is \(180^{\circ}\). Let's consider the fact that we can find the angle \(x\) using the angle - relationships. Since \(AB\parallel CD\), \(\angle BAC = 31^{\circ}\) (alternate - interior angles). In \(\triangle ABC\), we know that the angle opposite to the side \(AB\) and the angle \(x\) are related.
We know that in a parallelogram, opposite angles are equal and adjacent angles are supplementary. But we can also use the fact that in \(\triangle ABC\), if we consider the angle - sum property. Since \(AB\parallel CD\), the angle corresponding to the given \(31^{\circ}\) angle in the other part of the parallelogram formed by the diagonal is also \(31^{\circ}\). So \(x = 31^{\circ}\).

Answer:

A. 31