Question

given: abcd is a parallelogram. diagonals ac, bd intersected at e. prove ae ≅ ce and be ≅ de. assemble the proof by dragging tiles to the statements and reasons columns.

Explanation:

Step1: Recall properties of parallelogram

In parallelogram \(ABCD\), \(AB\parallel CD\) and \(AB = CD\) (opposite - sides of a parallelogram are parallel and equal), and \(\angle BAE=\angle DCE\) (alternate interior angles as \(AB\parallel CD\) and \(AC\) is a transversal), \(\angle ABE=\angle CDE\) (alternate interior angles as \(AB\parallel CD\) and \(BD\) is a transversal).

Step2: Prove triangle congruence

In \(\triangle ABE\) and \(\triangle CDE\), we have \(\angle BAE=\angle DCE\), \(AB = CD\), \(\angle ABE=\angle CDE\). So, \(\triangle ABE\cong\triangle CDE\) by the Angle - Side - Angle (ASA) congruence criterion.

Step3: Use congruent - triangle properties

Since \(\triangle ABE\cong\triangle CDE\), corresponding parts of congruent triangles are equal. So, \(AE = CE\) and \(BE=DE\) (corresponding sides of congruent triangles are equal).

Answer:

Statements:

1. \(ABCD\) is a parallelogram

• Reason: Given

1. \(AB\parallel CD\) and \(AB = CD\)

• Reason: Opposite sides of a parallelogram are parallel and equal

1. \(\angle BAE=\angle DCE\)

• Reason: Alternate interior angles ( \(AB\parallel CD\), \(AC\) is a transversal)

1. \(\angle ABE=\angle CDE\)

• Reason: Alternate interior angles ( \(AB\parallel CD\), \(BD\) is a transversal)

1. \(\triangle ABE\cong\triangle CDE\)

• Reason: ASA congruence criterion

1. \(AE = CE\) and \(BE = DE\)

• Reason: Corresponding sides of congruent triangles are equal