In the figure, KN and JL are straight lines. LPK is an equilateral triangle and JK // ML. The sum of $\angle MJL$ and $\angle LMJ$ is $138^\circ$. Find $\angle QLP$. Ans: ___ $^\circ$

1. Find $\angle MLJ$ using the sum of angles in $\triangle JML$.

In $\triangle JML$, the sum of angles is $180^\circ$.
$\angle MLJ = 180^\circ - (\angle MJL + \angle LMJ)$
$\angle MLJ = 180^\circ - 138^\circ = 42^\circ$

2. Find $\angle KJL$ using alternate interior angles.

Since $JK \parallel ML$, alternate interior angles are equal.
$\angle KJL = \angle MLJ = 42^\circ$

3. Find $\angle JLK$ using the sum of angles in right-angled $\triangle JKL$.

In $\triangle JKL$, $\angle JKL = 90^\circ$.
$\angle JLK = 180^\circ - 90^\circ - 42^\circ = 48^\circ$

4. Find $\angle QLP$ using the properties of the equilateral triangle.

Since $LPK$ is an equilateral triangle, $\angle PLK = 60^\circ$.
Since $J, Q, L$ are collinear, $\angle QLK = \angle JLK = 48^\circ$.
$\angle QLP = \angle PLK - \angle QLK$
$\angle QLP = 60^\circ - 48^\circ = 12^\circ$