Question

proving the parallelogram diagonal theorem
given: abcd is a parallelogram.
diagonals \\(\overline{ac}\\), \\(\overline{bd}\\) intersect at e.
prove: \\(\overline{ae} \cong \overline{ce}\\) and \\(\overline{be} \cong \overline{de}\\)
angles segments triangles statements reasons
asa cpctc
parallelogram side theorem alternate interior angles theorem
statements reasons

1. abcd is a parallelogram 1. given
2. \\(\overline{ab} \parallel \overline{cd}\\) 2. def. of parallelogram
3. \\(\angle bae\\) and \\(\angle dce\\) are alt. interior angles 3. def. of alt. interior angles
4. \\(\angle bae \cong \angle dce\\) 4. alternate interior angles theorem
5. \\(\angle abe\\) and \\(\angle cde\\) are alt. interior angles 5. def. of alt. interior angles

correct! assemble the next statement.

Explanation:

Step1: Prove congruent alternate angles

$\angle ABE \cong \angle CDE$

Step2: Cite congruent parallelogram sides

$\overline{AB} \cong \overline{CD}$ (parallelogram side theorem)

Step3: Apply ASA congruence

$\triangle ABE \cong \triangle CDE$ (ASA)

Step4: Use CPCTC for segment congruence

$\overline{AE} \cong \overline{CE}, \overline{BE} \cong \overline{DE}$ (CPCTC)

Answer:

Next Statement & Reason (to fill the blanks):

1. $\boldsymbol{\angle ABE \cong \angle CDE}$
2. alternate interior angles theorem

Full Completed Proof (final conclusion):

$\overline{AE} \cong \overline{CE}$ and $\overline{BE} \cong \overline{DE}$