Question

if \\(\overline{tw} \cong \overline{vw}\\), \\(m\angle tuw = 2s\\), and \\(m\angle vuw = s + 23^\circ\\), what is the value of \\(s\\)?
\\(s = \square^\circ\\)

Explanation:

Step1: Use the Angle Bisector Theorem

Since \( \overline{TW}\cong\overline{VW}\) and \(WU\) is the common side, and \( \angle V = \angle T=90^{\circ}\), by the Hypotenuse - Leg (HL) congruence criterion for right - triangles (\(\triangle VUW\) and \(\triangle TUW\)), we know that \(\triangle VUW\cong\triangle TUW\). Then, by the property of congruent triangles (corresponding angles are equal), \(m\angle T UW=m\angle VUW\).

Step2: Set up the equation

We are given that \(m\angle T UW = 2s\) and \(m\angle VUW=s + 23^{\circ}\). Since \(m\angle T UW=m\angle VUW\), we can set up the equation \(2s=s + 23^{\circ}\).

Step3: Solve the equation for \(s\)

Subtract \(s\) from both sides of the equation \(2s=s + 23^{\circ}\).

$$2s-s=s + 23^{\circ}-s$$

$$s = 23^{\circ}$$

Answer:

\(23\)