Answer
$m \angle K = 51^{\circ}$
Work Step by Step
First, we identify all the pairs of congruent angles:
$\angle D ≅ \angle H$
$\angle F ≅ \angle K$
$\angle G ≅ \angle M$
We see that $\angle F ≅ \angle K$, so $m \angle F = m \angle K$; however, we do not have the measure for either of these angles, so let us try to find the measure of $\angle F$.
We are given the measures of two angles in $\triangle DFG$, so we can use the triangle-sum theorem, which states that the sum of the interior angles of a triangle is equal to $180^{\circ}$, to find the measure of the last angle:
$m \angle F = 180 - (70 + 59)$
Evaluate what is in parentheses first:
$m \angle F = 180 - (129)$
Subtract to solve:
$m \angle F = 51$
If $m \angle F = 51^{\circ}$, then $m \angle K = 51^{\circ}$.
