Question

Given quadrilateral ABCD, where the diagonals AC and BD intersect at point E. AE⎯⎯⎯⎯⎯⎯⎯≅EC⎯⎯⎯⎯⎯⎯⎯⎯AE¯≅EC¯ and BE⎯⎯⎯⎯⎯⎯⎯≅DE⎯⎯⎯⎯⎯⎯⎯⎯BE¯≅DE¯. Can you prove can you prove that the figure is a parallelogram? Explain.

Answer 1

By utilizing the Side-Side-Side postulate to show congruence between triangles formed by the diagonals of the quadrilateral and then demonstrating that the opposite sides are congruent as a result, we can prove that quadrilateral ABCD must be a parallelogram.

Step-by-step explanation:

To prove that a quadrilateral is a parallelogram, we can use the property that states if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. In this case, we're given a quadrilateral ABCD with diagonals AC and BD intersecting at point E, and it's given that AE≈EC and BE≈DE. Because AE and EC are equal, as are BE and DE, it implies that triangles ABE and CDE are congruent (by the Side-Side-Side postulate). Therefore, we have corresponding parts of congruent triangles that are congruent, which means AB≈CD and AD≈BC. Thus, we have shown that both pairs of opposite sides of quadrilateral ABCD are congruent, which by definition makes ABCD a parallelogram.