In the diagram, a∥b and e∥f, forming a parallelogram. Write a proof that proves that opposite sides of the parallelogram are congruent. write a paragraph proof, but be sure to include a reason for each of your statements.

Answer 1

To prove that opposite sides of a parallelogram are congruent, consider parallelogram ABCD where \( AB \parallel CD \) and \( AD \parallel BC \). By the properties of parallel lines and transversal lines, we know that corresponding angles are equal. Thus, \(\angle DAB = \angle BCD\) and \(\angle ABC = \angle ADC\) (Reason: Alternate interior angles are congruent).

Next, we can apply the Side-Angle-Side (SAS) Congruence Postulate. In triangles \( \triangle DAB \) and \( \triangle CBA \), we have \( AD = BC \) (since they are opposite sides of the parallelogram), \(\angle DAB = \angle ABC\), and \( AB = CD \) (again, opposite sides of the parallelogram). Since two angles and the included side in both triangles are congruent, it follows that \(\triangle DAB \cong \triangle CBA\) (Reason: SAS Congruence Postulate).

This congruence indicates that the corresponding sides of these triangles, which are \( AD \) and \( BC \), are congruent, giving us \( AD \cong BC \). Similarly, by examining triangles \( \triangle ABC \) and \( \triangle CDA \), we can apply the same reasoning to find that \( AB \cong CD \). Thus, we conclude that opposite sides of the parallelogram are congruent: \( AB \cong CD \) and \( AD \cong BC \) (Reason: Corresponding parts of congruent triangles are congruent, or CPCTC). This completes the proof that opposite sides of a parallelogram are congruent.