Question

in △vwx, \\( \overline{vw}=\overline{vx} \\). if \\( m\angle v = 50^{\circ} \\), what are measures of \\( \angle w \\) and \\( \angle x \\)?
\\( \angle w = 65^{\circ} \\) and \\( \angle x = 65^{\circ} \\)
\\( \angle w = 50^{\circ} \\) and \\( \angle x = 80^{\circ} \\)
\\( \angle w = 80^{\circ} \\) and \\( \angle x = 50^{\circ} \\)
\\( \angle w = 30^{\circ} \\) and \\( \angle x = 100^{\circ} \\)

Explanation:

Step1: Identify the triangle type

Since \( \overline{VW}=\overline{VX} \), \( \triangle VWX \) is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are equal. So \( \angle W=\angle X \).

Step2: Use the angle - sum property of a triangle

The sum of the interior angles of a triangle is \( 180^{\circ} \). Let \( \angle W = \angle X=x \). We know that \( \angle V = 50^{\circ} \). Then, by the angle - sum property \( \angle V+\angle W+\angle X = 180^{\circ} \). Substitute the values: \( 50^{\circ}+x + x=180^{\circ} \).

Step3: Solve the equation for \( x \)

Combine like terms: \( 50^{\circ}+2x=180^{\circ} \). Subtract \( 50^{\circ} \) from both sides: \( 2x=180^{\circ}- 50^{\circ}=130^{\circ} \). Divide both sides by 2: \( x=\frac{130^{\circ}}{2}=65^{\circ} \).

Answer:

\(\angle W = 65^{\circ}\) and \(\angle X = 65^{\circ}\) (first option)