Question

given \\( \triangle lmn \cong \triangle pqr \\) set up and solve an equation to solve for x.

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle LMN\cong\triangle PQR\), their corresponding angles are equal. \(\angle N\) in \(\triangle LMN\) corresponds to \(\angle R\) in \(\triangle PQR\). But wait, no, we know that the sum of angles in a triangle is \(180^{\circ}\). Also, for congruent triangles, corresponding angles are equal. Wait, actually, if \(\triangle LMN\cong\triangle PQR\), then \(\angle M = \angle Q\)'s adjacent - supplementary angle? No, no. Wait, the correct approach: In \(\triangle LMN\), \(\angle M = 112^{\circ}\), in \(\triangle PQR\), \(\angle R=42^{\circ}\). Using the angle - sum property of a triangle (\(\angle L+\angle M+\angle N = 180^{\circ}\) and \(\angle P+\angle Q+\angle R = 180^{\circ}\)), and since \(\triangle LMN\cong\triangle PQR\), \(\angle N=\angle R\) is wrong. Wait, no! Wait, the problem is to solve for \(x\). The angle \((13x - 36)^{\circ}\) in \(\triangle LMN\) and \(\angle R = 42^{\circ}\) (because of congruence \(\triangle LMN\cong\triangle PQR\), corresponding angles: \(\angle N=\angle R\))
So we set up the equation \(13x-36 = 42\)

Step2: Solve the linear equation

Add \(36\) to both sides of the equation:
\(13x-36 + 36=42 + 36\)
\(13x=78\)
Divide both sides by \(13\):
\(x=\frac{78}{13}\)

Answer:

\(x = 6\)