Question

name: chris.v example 2 given parallelogram abcd, prove △aed≅△ceb. statement reason abcd is a parallelogram given ac is a bisector of bd opposite sides in parallelograms are congruent. is a bisector of ac line - segments are equal to themselves (reflexive property) ae = ce a bisector creates two congruent segments. ad = bc opposite sides in a parallelogram are △aed≅△ceb

Explanation:

Step1: Recall property of parallelogram

In parallelogram \(ABCD\), the diagonals \(AC\) and \(BD\) bisect each other. So, \(AC\) is a bisector of \(BD\) and \(BD\) is a bisector of \(AC\). The point of intersection \(E\) is the mid - point of both \(AC\) and \(BD\). So, \(E\) is a bisector of \(AC\).

Step2: Identify congruent segments

Since \(E\) is the mid - point of \(AC\), by the definition of a bisector, \(AE = CE\). Also, since \(E\) is the mid - point of \(BD\), \(DE=BE\).

Step3: Identify vertical angles

\(\angle AED\) and \(\angle CEB\) are vertical angles. By the vertical - angles theorem, vertical angles are congruent. So, \(\angle AED\cong\angle CEB\).

Step4: Apply SAS congruence criterion

In \(\triangle AED\) and \(\triangle CEB\), we have \(AE = CE\), \(\angle AED=\angle CEB\), and \(DE = BE\). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle AED\cong\triangle CEB\).

Statement	Reason
\(AC\) is a bisector of \(BD\)	Diagonals of a parallelogram bisect each other
\(E\) is a bisector of \(AC\)	Diagonals of a parallelogram bisect each other
\(AE = CE\)	A bisector creates two congruent segments
\(DE=BE\)	A bisector creates two congruent segments
\(\angle AED\cong\angle CEB\)	Vertical angles are congruent
\(\triangle AED\cong\triangle CEB\)	Side - Angle - Side (SAS) congruence criterion

Answer:

The proof is completed as shown in the above - filled table, where we use the properties of parallelogram diagonals, the definition of bisectors, vertical - angles theorem and the SAS congruence criterion to prove \(\triangle AED\cong\triangle CEB\).