Question

these figures are congruent. what is lm?
____ miles

Explanation:

Step1: Find missing angle in ΔSTU

The sum of angles in a triangle is $180^\circ$.
$\angle S = 180^\circ - 61^\circ - 41^\circ = 78^\circ$

Step2: Match congruent triangle parts

In ΔNLM, $\angle M = 78^\circ$, adjacent sides $NM=67$ mi, $NL=75$ mi. In ΔSTU, $\angle S=78^\circ$, adjacent sides $US=50$ mi, $ST=67$ mi. Congruent triangles have matching sides: $LM$ corresponds to $TU$? No, match angle-side pairs: $\angle M \cong \angle S$, $NM \cong ST=67$ mi, so $LM$ corresponds to $SU$? Wait no, correct correspondence: ΔNLM ≅ ΔUTS (angle-side: $\angle M=78^\circ=\angle S$, $NM=ST=67$ mi, $NL=UT=75$ mi). So $LM$ corresponds to $SU$? No, calculate side length via congruence: the side opposite the 78° angle in ΔSTU? Wait no, in ΔSTU, sides: $SU=50$, $ST=67$, angle at U=61, T=41, S=78. In ΔNLM, angle at M=78, sides NM=67, NL=75. So congruent triangles: ΔLMN ≅ ΔSUT, so $LM = SU$? No, wait, the side opposite 75 mi in ΔNLM is angle M=78, in ΔSTU, side opposite 75 mi would be angle S=78, so $UT=75$ mi, so $LM$ is equal to the side in ΔSTU opposite angle U=61? No, better: use congruence: corresponding sides: $NM=ST=67$, $\angle M=\angle S=78^\circ$, so $LM$ corresponds to $SU$? No, wait, the side we need is $LM$, which is the side in ΔNLM between $\angle L$ and $\angle M$. In ΔSTU, the side between $\angle T$ and $\angle S$ is $ST=67$, no. Wait, calculate the third side of ΔSTU using Law of Sines to confirm.

Step3: Law of Sines for ΔSTU

$\frac{TU}{\sin 78^\circ} = \frac{SU}{\sin 41^\circ}$
$TU = \frac{50 \times \sin 78^\circ}{\sin 41^\circ} \approx \frac{50 \times 0.9781}{0.6561} \approx 75$ mi, which matches $NL=75$ mi. Then $\frac{ST}{\sin 61^\circ} = \frac{SU}{\sin 41^\circ}$, $ST=\frac{50 \times \sin 61^\circ}{\sin 41^\circ} \approx \frac{50 \times 0.8746}{0.6561} \approx 67$ mi, which matches $NM=67$ mi. So the remaining side $LM$ corresponds to $SU$? No, wait ΔNLM has sides: $NL=75$, $NM=67$, $LM=?$ ΔSTU has sides $SU=50$, $ST=67$, $TU=75$. So congruent pairs: $NL=TU$, $NM=ST$, so $LM=SU=50$ mi.

Answer:

50 miles