Question

if m∠tvu = m∠tvw = 41°, tu = x + 50, and tw = 3x, what is the value of x? diagram shows right angles at w (∠vw t) and u (∠vut), with points v, w, t, u and segment tv.

Explanation:

Step1: Identify the theorem

Since \( \angle TVU = \angle TVW = 41^\circ \), and \( \angle VW T = \angle VUT = 90^\circ \) (right angles), by the Angle - Angle - Side (AAS) congruence criterion, \( \triangle TVW \cong \triangle TVU \). Therefore, the corresponding sides \( TU \) and \( TW \) are equal.

Step2: Set up the equation

We know that \( TU=x + 50 \) and \( TW = 3x \). Since \( TU=TW \) (corresponding sides of congruent triangles), we can set up the equation:
\( x + 50=3x \)

Step3: Solve for \( x \)

Subtract \( x \) from both sides of the equation:
\( 50=3x - x \)
\( 50 = 2x \)
Divide both sides by 2:
\( x=\frac{50}{2}=25 \)

Answer:

\( 25 \)