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 parallel lines and alternate - interior angles

Since \(\overleftrightarrow{URV}\parallel\overleftrightarrow{ST}\), we know that \(\angle SRT=\angle URS\) (alternate - interior angles). But we need to use the angle - sum property of a triangle. In \(\triangle RST\), the sum of the interior angles of a triangle is \(180^{\circ}\).

Step2: Apply the angle - sum formula for a triangle

The formula for the sum of interior angles of a triangle \(\triangle RST\) is \(\angle S+\angle T+\angle R = 180^{\circ}\). We are given \(\angle S = 50^{\circ}\) and \(\angle T=72^{\circ}\). Substitute these values into the formula: \(x + 50+72=180\).

Step3: Solve for \(x\)

Simplify the left - hand side of the equation: \(x+122 = 180\). Then subtract 122 from both sides: \(x=180 - 122\).

Answer:

\(x = 58\)