To prove that the opposite sides of a parallelogram are congruent using a transformational approach, we can apply vector addition and the parallelogram rule. By translating vectors in parallel, we can establish the congruency of opposite sides based on vector properties.
To prove that the opposite sides of a parallelogram are congruent using a transformational approach, we can apply the concept of vector addition and the parallelogram rule. Two vectors A and B, when placed such that their origins coincide, will form a parallelogram if we draw lines parallel to each vector at the terminal point of the other. The newly formed sides of the parallelogram are congruent to the original vectors A and B.
To visualize this, consider the parallelogram ABCD, where AB is vector A and AD is vector B. Vector AC, which represents the resultant R = A + B, will be the diagonal running from the origin to the opposite corner. By constructing another parallelogram with sides BC and CD parallel to DA and AB respectively, we can see that BC is congruent to DA and CD is congruent to AB because they are formed by parallel translation of the vectors. Hence, in the case of the parallelogram ABCD, opposite sides AB and CD are congruent, as are AD and BC.
This method presupposes the properties of vector addition and parallel translation, which state that vectors retain their magnitude and direction when moved in parallel. This also aligns with the definition and properties of a parallelogram, where opposite sides are parallel and, as proven by vectors, also congruent.