Question: The measure of AC = $7$, $m\angle BAC = 80^\circ$, $m\angle ABC = 50^\circ$, the measure of NM = $7$, $m\angle LNM = 80^\circ$, and $m\angle NLM = 50^\circ$.
Do the measurements indicate that $\triangle ABC \cong \triangle NLM$?
A. Yes, the measures of two sets of corresponding angles are equal.
B. No, the given angles are not corresponding.
C. Yes, the measures of two sets of corresponding angles on each triangle are equal and the measures of one set of corresponding non-included sides are equal.
D. No, the one set of corresponding sides are included in the given angles.
1. **State the problem:** We are given two triangles $\triangle ABC$ and $\triangle NLM$ with side lengths and angle measures:
- $AC = 7$, $m\angle BAC = 80^\circ$, $m\angle ABC = 50^\circ$
- $NM = 7$, $m\angle LNM = 80^\circ$, $m\angle NLM = 50^\circ$
We need to determine if $\triangle ABC \cong \triangle NLM$ based on these measurements.
2. **Recall the Angle-Side-Angle (ASA) Congruence Rule:**
Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of another triangle.
3. **Identify corresponding parts:**
- $\angle BAC$ corresponds to $\angle LNM$ (both $80^\circ$)
- $\angle ABC$ corresponds to $\angle NLM$ (both $50^\circ$)
- Side $AC$ corresponds to side $NM$ (both length $7$)
4. **Check if the side is included between the two angles:**
In $\triangle ABC$, side $AC$ is between $\angle BAC$ and $\angle ABC$?
- $\angle BAC$ is at vertex $A$, between sides $AB$ and $AC$.
- $\angle ABC$ is at vertex $B$, between sides $AB$ and $BC$.
Side $AC$ connects vertices $A$ and $C$, so it is adjacent to $\angle BAC$ but not adjacent to $\angle ABC$.
Therefore, side $AC$ is **not** the included side between the two given angles.
Similarly, in $\triangle NLM$, side $NM$ is between $\angle LNM$ and $\angle NLM$?
- $\angle LNM$ is at vertex $N$, between sides $NL$ and $NM$.
- $\angle NLM$ is at vertex $L$, between sides $NL$ and $LM$.
Side $NM$ connects vertices $N$ and $M$, so it is adjacent to $\angle LNM$ but not adjacent to $\angle NLM$.
Thus, side $NM$ is **not** the included side between the two given angles.
5. **Conclusion:**
Since the side given is not the included side between the two angles, the congruence cannot be established by ASA.
However, the two angles and a non-included side are equal, which corresponds to the Angle-Angle-Side (AAS) congruence criterion.
Therefore, the triangles are congruent by AAS.
6. **Answer choice:**
Option C: "Yes, the measures of two sets of corresponding angles on each triangle are equal and the measures of one set of corresponding non-included sides are equal."