Question

in the figure below, \\( \triangle rst \\) is drawn. the line \\( \overleftrightarrow{urv} \\) is drawn such that \\( \overleftrightarrow{urv} \parallel \overline{st} \\).

Explanation:

Step1: Use the property of alternate interior angles

Since \(\overleftrightarrow{URV}\parallel\overleftrightarrow{ST}\), \(\angle SRT=\angle RST\) (alternate - interior angles). Let's first find the third angle in \(\triangle RST\).
We know that the sum of angles in a triangle is \(180^{\circ}\).

Step2: Calculate the value of \(x\)

In \(\triangle RST\), \(\angle RTS = 72^{\circ}\), \(\angle RST=50^{\circ}\).
Using the angle - sum formula for a triangle \(\angle RTS+\angle RST+\angle SRT = 180^{\circ}\).
Substitute the known values: \(72^{\circ}+50^{\circ}+x^{\circ}=180^{\circ}\).
Simplify the left - hand side: \(122^{\circ}+x^{\circ}=180^{\circ}\).
Subtract \(122^{\circ}\) from both sides: \(x^{\circ}=180^{\circ}- 122^{\circ}\).

Answer:

\(x = 58\)