Question

$mangle efg = mangle feh$ because they are alternate interior angles.$mangle fge = mangle gei$ because they are alternate interior angles.$mangle feh + x^circ + mangle gei = 180^circ$ because the three angles form a straight line.so the value of $x$ must be 46after moving the vertices, the new value of $x$ is $\boldsymbol{^circ}$

Explanation:

Step1: Recall triangle angle sum

The sum of angles in $\triangle EFG$ is $180^\circ$.
$\angle EFG + \angle FGE + \angle GEF = 180^\circ$

Step2: Substitute known angles

Plug in $\angle EFG=91^\circ$, $\angle FGE=43^\circ$.
$91^\circ + 43^\circ + x^\circ = 180^\circ$

Step3: Calculate sum of known angles

Add the two given angles.
$134^\circ + x^\circ = 180^\circ$

Step4: Solve for $x$

Subtract $134^\circ$ from both sides.
$x^\circ = 180^\circ - 134^\circ = 46^\circ$

Step5: Verify with straight line rule

Angles at $E$ form a straight line: $\angle FEH + x^\circ + \angle GEI = 180^\circ$. Substitute $\angle FEH=91^\circ$, $\angle GEI=43^\circ$.
$91^\circ + x^\circ + 43^\circ = 180^\circ$
$x^\circ = 180^\circ - 134^\circ = 46^\circ$

Answer:

$46$

For the final blank about the new value of $x$ after moving vertices: Since the sum of angles in a triangle and properties of parallel lines (alternate interior angles) remain unchanged regardless of moving vertices while keeping the parallel lines and angle relationships intact, the new value of $x$ is still $46$.