Which of the following pairs of triangles can be proven congruent using the $\mathrm{SAS}$ (Side-Angle-Side) criterion rather than the Law of Sines?

• A. Two triangles with all three angles known to be equal in each triangle
• B. Two triangles with all three sides known to be equal in each triangle
• C. Two triangles with two angles and a non-included side known to be equal in each triangle
• D. Two triangles with two sides and the included angle known to be equal in each triangle

Answer: (D)

Verified step by step guidance

1. Recall the SAS (Side-Angle-Side) congruence criterion: two triangles are congruent if two sides and the included angle (the angle between those two sides) of one triangle are respectively equal to two sides and the included angle of the other triangle.
2. Analyze each option to see if it fits the SAS criterion:
3. Option 1: Triangles with all three angles equal. This is Angle-Angle-Angle (AAA), which shows similarity but not congruence, so SAS does not apply here.
4. Option 2: Triangles with all three sides equal. This is Side-Side-Side (SSS) congruence, which is a different criterion from SAS.
5. Option 3: Triangles with two angles and a non-included side equal. This is Angle-Angle-Side (AAS), which is not SAS because the side is not between the two angles.
6. Option 4: Triangles with two sides and the included angle equal. This matches the SAS criterion exactly, so these triangles can be proven congruent using SAS.