For this proof, we need to complete the missing statements and reasons to demonstrate that the diagonals of parallelogram $ABCD$ bisect each other. Here’s the reasoning to fill in the blanks:

1. Given: $AB \parallel CD$ and $AD \parallel BC$
2. Reason: Given.
3. Given: $\angle 1 \cong \angle 3$
4. Reason: When a transversal crosses parallel lines, alternate interior angles are congruent.
5. Given: $\angle 2 \cong \angle 4$
6. Reason: When a transversal crosses parallel lines, alternate interior angles are congruent.

The steps to fill in for each blank:

• Step 5 (Pick statement): $\triangle ABE \cong \triangle CDE$ Reason: Pick congruence criterion ASA (Angle-Side-Angle)

• Step 6 (Pick statement): $AE \cong CE$ and $BE \cong DE$ Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Let me know if you want more details on each step or have any other questions!

Related Questions

1. How do you identify congruent triangles in a parallelogram?
2. What is the significance of alternate interior angles in parallel lines?
3. How does the CPCTC theorem help in geometric proofs?
4. What other methods could prove that diagonals bisect in a parallelogram?
5. Why is the ASA congruence criterion applicable in this proof?

Tip

Remember, identifying congruent parts of triangles can help conclude that segments are equal, a key step in proving bisectors.