Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Which theorem would you refer to in proving triangles are congruent based on two sides and the included angle?

1. AAS

2. SSS

3. SAS

4. ASA

The correct answer is: SAS

The theorem you would refer to for proving triangles are congruent based on two sides and the included angle is the Side-Angle-Side (SAS) Congruence Theorem. This theorem states that if two sides of one triangle are equal in length to two sides of another triangle, and the angle formed between those two sides is also equal, then the triangles are congruent. In this context, the focus is on two sides and the included angle, which means the angle is the one that is directly between the two sides you are considering for congruence. This makes SAS particularly useful because it combines side lengths and the angle measurement to establish congruence, formulating a definitive case based on the geometric properties involved. The other options refer to different criteria for triangle congruence: AAS (Angle-Angle-Side) involves two angles and a non-included side, SSS (Side-Side-Side) requires all three sides to be equal, and ASA (Angle-Side-Angle) requires two angles and the included side to establish congruence. Each of these has distinct criteria that do not apply to the situation described, where the emphasis is on two sides and the angle formed between them.