Question

$mangle stu = mangle tsv$ because they are alternate interior angles.$mangle tus = mangle usw$ because they are alternate interior angles.$mangle tsv + x^circ + mangle usw = 180^circ$ because the three angles form a straight line.so the value of $x$ must be 60after moving the vertices, the new value of $x$ is

Explanation:

Step1: Recall triangle angle sum rule

The sum of angles in a triangle is $180^\circ$. For $\triangle STU$, $m\angle STU + m\angle TUS + x^\circ = 180^\circ$.

Step2: Substitute known angle values

We know $m\angle STU=63^\circ$, $m\angle TUS=41^\circ$.
$63^\circ + 41^\circ + x^\circ = 180^\circ$

Step3: Calculate sum of known angles

$63+41=104$, so $104^\circ + x^\circ = 180^\circ$

Step4: Solve for $x$

$x^\circ = 180^\circ - 104^\circ$

When vertices of a triangle are moved, the measures of the corresponding alternate interior angles remain equal to the triangle's interior angles. The sum of angles on a straight line is still $180^\circ$, so the relationship $m\angle TSV + x^\circ + m\angle USW = 180^\circ$ holds, with $m\angle TSV=m\angle STU$ and $m\angle USW=m\angle TUS$. Since the sum of a triangle's interior angles is always $180^\circ$, $x$ will always equal $180^\circ - (m\angle STU + m\angle TUS)$, so its value does not change.

Answer:

$76^\circ$

For the follow-up question about moving vertices: