Answer: A. $22^\circ$

Explanation:1. Recall the sum of angles in a triangle: The sum of the interior angles of any triangle is $180^{\circ}$. So, $m\angle J + m\angle K + m\angle L = 180^{\circ}$.

2. Use the given information: We are given that $m\angle J = m\angle K$. Let's call this measure $x$. So, $m\angle J = x$ and $m\angle K = x$. We are given $m\angle L = 136^{\circ}$.

3. Substitute the values into the sum of angles equation: $x + x + 136^{\circ} = 180^{\circ}$ $2x + 136^{\circ} = 180^{\circ}$

4. Solve for x (which is the measure of $\angle J$): Subtract $136^{\circ}$ from both sides: $2x = 180^{\circ} - 136^{\circ}$ $2x = 44^{\circ}$ Divide by 2: $x = \dfrac{44^{\circ}}{2}$ $x = 22^{\circ}$

5. State the measure of $\angle J$: Since $m\angle J = x$, the measure of $\angle J$ is $22^{\circ}$.

6. Compare with the given options: This matches option [A].

Conclusion:The measure of $\angle J$ is $22^{\circ}$.