Final answer:
The parallelogram side theorem can be proven using the side-angle-side (SAS) congruence criterion by showing that if one pair of opposite sides in a quadrilateral are both congruent and parallel, the quadrilateral is a parallelogram.
Step-by-step explanation:
The question asks how to prove the parallelogram side theorem, which can be proven using congruence criteria. The correct option to prove this theorem is b) By using the side-angle-side (SAS) congruence criterion. This is because if one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral must be a parallelogram. By using SAS, we establish that one pair of triangles within the quadrilateral are congruent, and thus their corresponding sides (opposite sides of the parallelogram) are congruent.
To illustrate, consider a parallelogram ABCD with AB parallel to CD and AB congruent to CD. By the properties of a parallelogram, angle A is congruent to angle C. Additionally, AD is congruent to BC because opposite sides of a parallelogram are congruent. Thereby, triangle ABD and triangle CDB can be proved congruent using the SAS Postulate since they have two sides and the included angle congruent. Hence, we've proved that the two sides of the parallelogram are congruent, which is the parallelogram side theorem. In conclusion, no matter the principles of physics being applied, whether in geometry or vectors, the consistent result aligns with established mathematical theorems such as the Pythagorean theorem.