Answer
step statement Reason
2 BD bisect AC Diagonals of a parallelogram bisects each other
3 AE = EC A segment bisector divides a segment into two congruent segments
4 BE = DE A segment bisector divides a segment into two congruent segments
5 BE = BE A segment is congruent to itself
6 ΔABE = ΔCBE The three sides are congruent
7 ∠BEA = ∠BEC Corresponding parts of congruent triangles are congruent (CPCTC)
8 ∠BEA and ∠BEC are supplementary If two angles form a linear pair then they are supplementary.
9 ∠BEA = 90° ∠BEA = ∠BEC and they are supplementary
10 AC ⊥ BD They form an angle of 90°
Step-by-step explanation:
Since ABCD is a parallelogram, the diagonals bisect each other, so, we can start the prove as:
step statement Reason
2 BD bisect AC Diagonals of a parallelogram bisects each other
Then, AE = EC and BE = DE, so
step: 3
statement: AE = EC
Reason: A segment bisector divides a segment into two congruent segments
Step: 4
Statement: BE = DE
Reason: A segment bisector divides a segment into two congruent segments
Now, since AB = BC, AE = EC, and BE = BE, the triangles ABE and ACE are congruent, so:
step statement Reason
5 BE = BE A segment is congruent to itself
6 ΔABE = ΔCBE The three sides are congruent
Finally, if the triangles are congruent the corresponding angles are also congruent, so:
step statement Reason
7 ∠BEA = ∠BEC Corresponding parts of congruent triangles are congruent (CPCTC)
Additionally, these angles are supplementary because they form a straight line, so:
step: 8
statement: ∠BEA and ∠BEC are supplementary
Reason: If two angles form a linear pair then they are supplementary.
Therefore, if they sum 180 degrees and they are equal, the only option is that each one measures 90 degrees. So,
Step statement Reason
9 ∠BEA = 90° ∠BEA = ∠BEC and they are supplementary
10 AC ⊥ BD They form an angle of 90°