We are given an isosceles trapezoid $MLKN$ and we need to find the measure of angle $K$. We are given that the measure of angle $L$ is $105^\circ$.

1. Problem Description

We are given an isosceles trapezoid $MLKN$ and we need to find the measure of angle $K$. We are given that the measure of angle $L$ is $105^\circ$.

2. Solution Steps

In an isosceles trapezoid, the base angles are equal. This means that the angles at the ends of each parallel side are equal. In this case, $MN$ and $LK$ are the non-parallel sides. Since the trapezoid is isosceles, angle $L$ is equal to angle $K$.

Therefore, if angle $L$ is $105^\circ$, then angle $K$ is also $105^\circ$.

3. Final Answer

The measure of angle $K$ is $105^\circ$.