#### Answer

$m \angle JKM = 35$

#### Work Step by Step

If the two legs of a triangle are congruent, then that triangle is an isosceles triangle. With isosceles triangles, we need to remember that the base angles are congruent. Also, since $\angle JKM$ and $\angle LKM$ are congruent, that means that $\overline{KM}$ bisects $\overline{JL}$ perpendicular to $\overline{JL}$. This means that $m \angle JMK$ and $m \angle L
MK$ are both $90^{\circ}$.
According to the triangle sum theorem, the measures of all the interior angles of a triangle equal $180^{\circ}$. If $m \angle J$ is $55$ and $m \angle JMK$ is $90$, then we can find the measure of the last angle. Let's set up that equation:
$m \angle JKM + m \angle JMK + m \angle J = 180$
Let's substitute in what we know:
$m \angle JKM + 90 + 55 = 180$
Add the constants on the left side of the equation:
$m \angle JKM + 145 = 180$
Subtract $145$ from both sides of the equation to solve for $m \angle J$:
$m \angle JKM = 35$