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Question

In the diagram, a∥b and e∥f , forming a parallelogram. Write a proof that proves that opposite sides of the parallelogram are congruent. You can write a two-column proof or a paragraph proof, but be sure to include a reason for each of your statements.(4 points)

Answer 1

Paragraph Proof

To prove that opposite sides of the parallelogram are congruent, we start with the definition of a parallelogram. A parallelogram is defined as a quadrilateral with both pairs of opposite sides parallel. In this case, we know that \( a \parallel b \) and \( e \parallel f \).

1. By definition of a parallelogram, opposite sides are parallel: \( a \parallel b \) and \( e \parallel f \).
2. Since \( a \parallel b \), we can assert that alternate interior angles formed by a transversal cutting across lines \( a \) and \( b \) are equal. This means that angle \( 1 \) (formed with side \( e \)) is equal to angle \( 2 \) (formed with side \( f \)). \( \angle 1 = \angle 2 \).
3. By a similar argument, we can apply the same reasoning to sides \( e \) and \( f \) with \( a \) and \( b \) providing alternate interior angles that are congruent.
4. Consequently, we can conclude that angles opposite to equal angles in triangles formed within the parallelogram are also equal; thus, triangles formed by the diagonals of the parallelogram must be congruent by the Angle-Angle (AA) criterion.
5. By the properties of congruent triangles, we know that corresponding sides of congruent triangles are congruent. Therefore, the lengths of opposite sides of the parallelogram must be equal: \( a = b \) and \( e = f \).

Thus, the opposite sides of the parallelogram are proven to be congruent.

Two-Column Proof

| Statement | Reason | |---------------------------------------------------------------------------|---------------------------------------------------------------------------------------------| | 1. \( a \parallel b \) and \( e \parallel f \) | Given (the definition of the parallelogram) | | 2. \( \angle 1 = \angle 2 \) (alternate interior angles) | If two parallel lines are cut by a transversal, then alternate interior angles are equal. | | 3. Similar reasoning applies for angles along other lines | Alternate interior angles with transversal across lines \( e \), \( f \) are also equal. | | 4. Triangles formed by the diagonals of the parallelogram are congruent. | Two pairs of angles (by AA criteria) confirm congruence of the triangles. | | 5. Corresponding sides of congruent triangles are congruent. | A property of congruent triangles. | | 6. Thus, \( a = b \) and \( e = f \) | Opposite sides of a parallelogram are congruent. |

This establishes the necessary proof that the opposite sides of a parallelogram are congruent.