Question

$mangle hlj = mangle ihk$ because they are alternate interior angles.$mangle ljh = mangle jhl$ because they are alternate interior angles.$mangle ihk + x^circ + mangle jhl = 180^circ$ because the three angles form a straight line.so the value of $x$ must be 83after moving the vertices, the new value of $x$ is

Explanation:

Step1: Use straight line angle sum

The angles on straight line $KL$ sum to $180^\circ$.
$79^\circ + x^\circ + 61^\circ = 180^\circ$

Step2: Calculate sum of known angles

Add the given angle values.
$79 + 61 = 140$

Step3: Solve for $x$

Subtract the sum from $180^\circ$.
$x = 180 - 140$

Answer:

$40$

For the follow-up question about moving vertices: Since the lines $KL$ and $IJ$ remain parallel, the alternate interior angle relationship holds, so the sum of angles on the straight line at $H$ will still be $180^\circ$. As long as the alternate interior angles match the angles at $I$ and $J$, the value of $x$ will still be calculated as $180^\circ$ minus the sum of the other two angles on the straight line, maintaining the same logical relationship. If the angles at $I$ and $J$ do not change, $x$ remains $40$. If the angles at $I$ and $J$ change, repeat the steps above with the new angle values.