1. **State the problem:** We have an isosceles triangle $\triangle LMN$ with base $NL$. The angles are given as $m \angle N = (2x + 33)^\circ$ and $m \angle L = (3x + 24)^\circ$. We need to find the degree measure of each angle in the triangle. 2. **Identify the properties:** Since $\triangle LMN$ is isosceles with base $NL$, the two sides $LM$ and $MN$ are equal. Therefore, the angles opposite these sides are equal. The base angles are $\angle N$ and $\angle L$, so $m \angle N = m \angle L$. 3. **Set up the equation:** Since $m \angle N = m \angle L$, we have: $$2x + 33 = 3x + 24$$ 4. **Solve for $x$:** $$2x + 33 = 3x + 24$$ $$33 - 24 = 3x - 2x$$ $$9 = x$$ 5. **Find the angles:** Substitute $x = 9$ into the expressions for $\angle N$ and $\angle L$: $$m \angle N = 2(9) + 33 = 18 + 33 = 51^\circ$$ $$m \angle L = 3(9) + 24 = 27 + 24 = 51^\circ$$ 6. **Find the third angle $\angle M$:** The sum of angles in a triangle is $180^\circ$: $$m \angle M = 180 - (m \angle N + m \angle L) = 180 - (51 + 51) = 180 - 102 = 78^\circ$$ **Final answer:** $m \angle L = 51^\circ$ $m \angle M = 78^\circ$ $m \angle N = 51^\circ$