Question

1. given that ae is congruent ce, de is congruent to be, and ad is parallel to bc. prove that quadrilateral abcd is a parallelogram

Explanation:

Step1: Prove triangles congruent

Since \(AD\parallel BC\), \(\angle ADE=\angle CBE\) (alternate - interior angles).
Given \(AE = CE\) and \(DE = BE\).
By the Side - Angle - Side (SAS) congruence criterion, \(\triangle ADE\cong\triangle CBE\).

Step2: Get \(AD = BC\)

From \(\triangle ADE\cong\triangle CBE\), we have \(AD = BC\) (corresponding parts of congruent triangles).

Step3: Use parallelogram property

We know that \(AD\parallel BC\) (given) and we just proved \(AD = BC\).
A quadrilateral with one pair of opposite sides parallel and equal is a parallelogram.

Answer:

Quadrilateral \(ABCD\) is a parallelogram because it has one pair of opposite sides (\(AD\) and \(BC\)) that are both parallel (given \(AD\parallel BC\)) and equal (proven from \(\triangle ADE\cong\triangle CBE\)).