Proving the Parallelogram Diagonal Theorem

Given ABCD is a parralelogam, Diagnals AC and BD intersect at E

Prove AE is conruent to CE and BE is congruent to DE

Step-by-step explanation:

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

The proof is given below.

Step-by-step explanation:

Given a parallelogram ABCD. Diagonals AC and BD intersect at E. We have to prove that AE is congruent to CE and BE is congruent to DE i. e diagonals of parallelogram bisect each other.

In ΔACD and ΔBEC

AD=BC (∵Opposite sides of parallelogram are equal)

∠DAC=∠BCE (∵Alternate angles)

∠ADC=∠CBE (∵Alternate angles)

By ASA rule, ΔACD≅ΔBEC

By CPCT (Corresponding Parts of Congruent triangles)

AE=EC and DE=EB

Hence, AE is conruent to CE and BE is congruent to DE