1. **State the problem:** Prove that point E bisects both diagonals AC and BD in a parallelogram ABCD. 2. **Given:** AB || CD and AD || BC (definition of a parallelogram). 3. **Use the theorem:** Opposite sides of a parallelogram are congruent, so AB \cong CD and AD \cong BC. 4. **Angles:** When a transversal crosses parallel lines, alternate interior angles are congruent, so \angle 1 \cong \angle 3 and \angle 2 \cong \angle 4. 5. **Triangles:** Using the congruence criteria (ASA or AAS), triangles formed by the diagonals and sides are congruent. 6. **Corresponding parts:** By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), segments AE \cong EC and BE \cong ED. 7. **Definition of bisector:** Since E divides both diagonals into two equal parts, E bisects AC and BD. **Final conclusion:** Point E is the midpoint of both diagonals AC and BD, proving the diagonals bisect each other in a parallelogram.