Which pair of triangles can be proven congruent by the $SAS$ (Side-Angle-Side) Congruence Theorem?

• A. Two triangles with two sides and the included $\mathrm{angle}$ equal in each triangle
• B. Two triangles with two $\mathrm{angles}$ and a non-included $\mathrm{side}$ equal in each triangle
• C. Two triangles with all three $\mathrm{sides}$ equal in each triangle
• D. Two triangles with all three $\mathrm{angles}$ equal in each triangle

Answer: (A)

Verified step by step guidance

1. Recall the SAS (Side-Angle-Side) Congruence Theorem states that two triangles are congruent if two sides and the included angle (the angle between those two sides) in one triangle are congruent to two sides and the included angle in another triangle.
2. Identify what 'included angle' means: it is the angle formed between the two given sides, not an angle that is separate or opposite to them.
3. Examine each option to see if it matches the SAS criteria:
4. - Option 1: Two triangles with two sides and the included angle equal in each triangle. This matches the SAS theorem exactly.
5. - Other options involve either two angles and a non-included side, all three sides, or all three angles, which correspond to different congruence criteria (ASA, SSS, AAA) and not SAS.