Question

Given ABCD is a parallelagram diagonals AC and BD intersect at E Prove AE is congruent to CE and BE is congruent to DE

Answer 1

In the parallelogram ABCD, diagonals AC and BD intersect at point E.

According to the definition of parallelogram, opposite sides are equal and parallel to each other. That means, AB = DC

Now as AB and DC are parallel, so according to the property of Alternate Interior Angles, we will get:

∠EAB = ∠ECD and ∠EBA = ∠EDC

Thus , in two triangles ΔABE and ΔDCE, two angles and one side are equal. So, ΔABE and ΔDCE are congruent to each other.

That means, AE = CE and BE = DE

So, AE is congruent to CE and BE is congruent to DE

Answer 2

1.  ABCD is a parallelogram --Given

2. AB≌CD--parallelogram side theorem

3. AB∥CD--def. of parellelogram

4. ∠ABE and ∠CDE are alt. interior angles-- def. of alt. interior angles

5.∠BAE and ∠DCE are alt. interior angles-- def. of alt. interior angles

6. ∠BAE≌∠DCE--alt. interior angles theorem

7. ∠ABE≌CDE--alt. interior angle theorm

8. ⊿BAE≌⊿DCE-- ASA

9. AE≌CE-- CPCTC

10. BE≌DE-- CPCTC