Which of the following pairs of triangles are congruent by the Angle-Angle-Side ($AAS$) criterion according to the Law of Sines ($\frac{a}{sin}=\frac{b}{sin}=\frac{c}{sin}$)?

• A. Triangle 1: $A=50°$, $B=60°$, $a=7$; Triangle 2: $A=50°$, $B=60°$, $b=7$
• B. Triangle 1: $A=30°$, $B=80°$, $a=10$; Triangle 2: $A=30°$, $B=80°$, $c=10$
• C. Triangle 1: $A=40°$, $B=70°$, $c=8$; Triangle 2: $A=40°$, $B=70°$, $c=8$
• D.

Answer: (C)

Verified step by step guidance

1. Recall that the Angle-Angle-Side (AAS) congruence criterion states that two triangles are congruent if two angles and a non-included side of one triangle are respectively equal to two angles and the corresponding non-included side of another triangle.
2. For each pair of triangles, first verify that the two given angles are equal in both triangles. Since the sum of angles in a triangle is 180°, the third angle can be found by subtracting the sum of the two given angles from 180°.
3. Next, identify which side is given in each triangle and check if it corresponds to the same relative position with respect to the two given angles. The side must be the one not included between the two angles (i.e., adjacent to only one of the given angles) for the AAS criterion to apply.
4. Use the Law of Sines, which states \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), to check if the given side lengths correspond correctly to their opposite angles in both triangles. This helps confirm if the side lengths are consistent with the given angles.
5. If both triangles have two equal angles and the corresponding non-included side equal, then by the AAS criterion, the triangles are congruent. Otherwise, they are not congruent by AAS.