Which theorem states that three sides of one triangle congruent to three sides of another triangle implies the triangles are congruent?

Which theorem states that three sides of one triangle congruent to three sides of another triangle implies the triangles are congruent?

• A. SAS
• B. ASA
• C. AAS
• D. SSS

Answer: (D)

The main idea is that knowing all three corresponding sides of two triangles are equal locks the triangles into a fixed shape and size. This is the SSS congruence criterion. If AB corresponds to DE, BC to EF, and CA to FD with all three pairs equal, there exists a rigid motion (a combination of rotation, reflection, and translation) that maps one triangle onto the other. Once one side, then the next, and finally the third side line up exactly, so every corresponding angle matches as well. In other words, three side equalities determine the triangle completely, so the two triangles are congruent. This is why three sides alone are enough, whereas the other common criteria rely on including an angle or two angles along with sides.

The main idea is that knowing all three corresponding sides of two triangles are equal locks the triangles into a fixed shape and size. This is the SSS congruence criterion. If AB corresponds to DE, BC to EF, and CA to FD with all three pairs equal, there exists a rigid motion (a combination of rotation, reflection, and translation) that maps one triangle onto the other. Once one side, then the next, and finally the third side line up exactly, so every corresponding angle matches as well. In other words, three side equalities determine the triangle completely, so the two triangles are congruent. This is why three sides alone are enough, whereas the other common criteria rely on including an angle or two angles along with sides.