Question

given: abcd is a parallelogram and \\(\overline{ab} \cong \overline{ce}\\).
prove: \\(\angle a \cong \angle e\\).

step | statement | reason
1 | abcd is a parallelogram; \\(\overline{ab} \cong \overline{ce}\\) | given
2 | \\(\overline{cd} \cong \overline{ab}\\) | opposite sides of a parallelogram / rhombus / rectangle / square are congruent
3 | \\(\overline{cd} \cong \overline{ce}\\) | transitive property
4 | \\(\angle e \cong \angle cde\\) | in a triangle, angles opposite of congruent sides are congruent
5 | \\(\overline{ab} \parallel \overline{cd}\\) | select a reason...

image of parallelogram abcd and triangle cde with markings

Explanation:

Step 1: Recall parallelogram property

In a parallelogram, opposite sides are parallel. So for parallelogram \(ABCD\), \(\overline{AB} \parallel \overline{CD}\) because of the definition of a parallelogram (opposite sides of a parallelogram are parallel).

Step 2: Determine the reason for step 5

The statement in step 5 is \(\overline{AB} \parallel \overline{CD}\), and the reason comes from the property of a parallelogram: Opposite sides of a parallelogram are parallel.

Step 3: Continue the proof (optional, but to show the flow)

After step 5, since \(\overline{AB} \parallel \overline{CD}\), \(\angle A\) and \(\angle CDE\) are corresponding angles (because \(AE\) is a transversal cutting the parallel lines \(AB\) and \(CD\)), so \(\angle A \cong \angle CDE\) (Corresponding Angles Postulate). Then from step 4, \(\angle E \cong \angle CDE\), so by the Transitive Property, \(\angle A \cong \angle E\). But for step 5, the reason is about the parallel sides of the parallelogram.

Answer:

The reason for step 5 (statement \(\overline{AB} \parallel \overline{CD}\)) is "Opposite sides of a parallelogram are parallel".