Question

solve for x.
options: 82.5, 180, 115, 55

Explanation:

Step1: Identify the figure type

The figure is a parallelogram (since opposite sides are parallel, indicated by the arrows). In a parallelogram, consecutive angles are supplementary (their sum is \(180^\circ\)). So, \(\angle B\) and \(\angle C\) are consecutive angles? Wait, no, looking at the sides: \(AB\) and \(DC\) are parallel (arrows), \(AD\) and \(BC\) are parallel. So \(\angle B\) and \(\angle C\): wait, actually, in a parallelogram, consecutive angles (adjacent angles) are supplementary. Wait, \(\angle B\) and \(\angle C\): let's check the sides. \(BC\) is parallel to \(AD\), and \(AB\) is parallel to \(DC\). So \(\angle B\) and \(\angle C\): are they same - side interior angles? Wait, no, \(\angle B\) is at vertex \(B\), between \(AB\) and \(BC\), and \(\angle C\) is at vertex \(C\), between \(BC\) and \(DC\). Since \(AB \parallel DC\) and \(BC\) is a transversal, \(\angle B\) and \(\angle C\) are same - side interior angles, so they should be supplementary. Wait, no, in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. Wait, maybe I made a mistake. Let's re - examine: in parallelogram \(ABCD\), \(AD\parallel BC\) and \(AB\parallel DC\). So \(\angle B\) and \(\angle A\) are consecutive, \(\angle B\) and \(\angle C\): wait, \(AB\parallel DC\), so \(\angle B + \angle C=180^\circ\)? Wait, no, if \(AB\parallel DC\) and \(BC\) is the transversal, then \(\angle B\) and \(\angle C\) are same - side interior angles, so they are supplementary. So \((2x + 15)+x=180\).

Step2: Solve the equation

We have the equation \(2x + 15+x=180\). Combine like terms: \(3x+15 = 180\). Subtract 15 from both sides: \(3x=180 - 15=165\). Then divide both sides by 3: \(x=\frac{165}{3}=55\)? Wait, no, wait, that gives \(x = 55\), but let's check again. Wait, maybe I mixed up the angles. Wait, in a parallelogram, opposite angles are equal, and consecutive angles (adjacent) are supplementary. Wait, \(\angle B\) and \(\angle D\) are equal? No, wait, \(\angle A=\angle C\), \(\angle B=\angle D\), and \(\angle A+\angle B = 180^\circ\). Wait, maybe I misidentified the angles. Let's look at the diagram again: \(\angle B=(2x + 15)^\circ\), \(\angle C=x^\circ\). If \(AB\parallel DC\) and \(BC\) is the transversal, then \(\angle B\) and \(\angle C\) are same - side interior angles, so they should be supplementary. So \((2x + 15)+x=180\). So \(3x+15 = 180\), \(3x=165\), \(x = 55\). Wait, but let's check the answer options. 55 is one of the options.

Answer:

\(x = 55\) (corresponding to the option "55")