Question

example 2 given parallelogram abcd, prove △aed≅△ceb. statement reason abcd is a __________ given ac is a bisector of ______ a bisector of ac a bisector creates two congruent segments.

Explanation:

Step1: Recall parallelogram property

In a parallelogram, the diagonals bisect each other.

Step2: Fill in the first blank

Since it is given that \(ABCD\) is a parallelogram, the first - blank should be filled with "parallelogram".

Step3: Recall diagonal - bisecting relationship

The diagonals of a parallelogram \(ABCD\) are \(AC\) and \(BD\). The point \(E\) is the intersection of the diagonals. So, \(AC\) is a bisector of \(BD\) (and vice - versa). The second blank should be filled with " \(BD\)".

Step4: Recall the property of the intersection of diagonals

The point \(E\) is the mid - point of both \(AC\) and \(BD\). So, \(BE = DE\) and \(AE=CE\) because a bisector creates two congruent segments.

Step5: Prove triangle congruence

In \(\triangle AED\) and \(\triangle CEB\), we have \(AE = CE\) (diagonals of parallelogram bisect each other), \(\angle AED=\angle CEB\) (vertically opposite angles are equal), and \(DE = BE\) (diagonals of parallelogram bisect each other). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle AED\cong\triangle CEB\).

Answer:

1. parallelogram
2. \(BD\)
3. \(E\) is a bisector of \(AC\)
4. \(AE = CE\), \(DE = BE\) (from the property of diagonals of a parallelogram bisecting each other)
5. \(\angle AED=\angle CEB\) (vertically opposite angles)
6. \(\triangle AED\cong\triangle CEB\) (by SAS congruence criterion)