Question

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}\\) | opposite sides of a parallelogram / rhombus / rectangle / square are parallel
6 | \\(\angle a \cong \angle cde\\) | select a reason...
(image of parallelogram abcd and triangle cde with markings)

Explanation:

Step1: Analyze the given information

We know that \( \overline{AB} \parallel \overline{CD} \) (from step 5, opposite sides of a parallelogram are parallel) and we have angles \( \angle A \) and \( \angle CDE \). Since \( \overline{AB} \parallel \overline{CD} \) and \( AE \) is a transversal, \( \angle A \) and \( \angle CDE \) are corresponding angles.

Step2: Recall the Corresponding Angles Postulate

The Corresponding Angles Postulate states that if two parallel lines are cut by a transversal, then the corresponding angles are congruent. So, because \( \overline{AB} \parallel \overline{CD} \) and \( AE \) is the transversal, \( \angle A \cong \angle CDE \) by the Corresponding Angles Postulate. Also, we can use the transitive property with the previous angle congruence (\( \angle E \cong \angle CDE \) from step 4) to get \( \angle A \cong \angle E \), but for step 6, the direct reason from the parallel lines and transversal is the Corresponding Angles Postulate. Alternatively, since we know \( \angle E \cong \angle CDE \) (step 4) and we can show \( \angle A \cong \angle CDE \) via corresponding angles, then by transitive property \( \angle A \cong \angle E \), but in the context of step 6, the most direct reason is Corresponding Angles Postulate (or Alternate Interior Angles, but Corresponding is more direct here). Another way: Since \( \overline{AB} \parallel \overline{CD} \), \( \angle A \) and \( \angle CDE \) are corresponding angles, so they are congruent. Also, from step 4, \( \angle E \cong \angle CDE \), so by transitive property \( \angle A \cong \angle E \), but for the reason of \( \angle A \cong \angle CDE \), it's Corresponding Angles Postulate.

Answer:

The reason for \( \angle A \cong \angle CDE \) is "Corresponding Angles Postulate (If two parallel lines are cut by a transversal, corresponding angles are congruent)" (or also can be related to the transitive property with \( \angle E \cong \angle CDE \), but the direct geometric reason from the parallel lines is the Corresponding Angles Postulate). If we consider the transitive property with \( \angle E \cong \angle CDE \) (step 4) and \( \angle A \cong \angle CDE \) (from parallel lines), then transitive property can also be used, but the more direct geometric reason is Corresponding Angles Postulate. However, looking at the flow, since \( \overline{AB} \parallel \overline{CD} \) and \( AE \) is the transversal, \( \angle A \) and \( \angle CDE \) are corresponding angles, so the reason is "Corresponding Angles Postulate" (or "If two parallel lines are cut by a transversal, corresponding angles are congruent") or also using the transitive property with \( \angle E \cong \angle CDE \) (step 4) and the fact that \( \angle A \) and \( \angle CDE \) are congruent (from parallel lines), but the key is that \( \angle A \) and \( \angle CDE \) are corresponding angles due to \( \overline{AB} \parallel \overline{CD} \), so the reason is "Corresponding Angles Postulate" (or "Alternate Interior Angles" but Corresponding is more accurate here as \( \angle A \) and \( \angle CDE \) are in corresponding positions relative to the transversal \( AE \) and parallel lines \( AB \) and \( CD \)).

Final answer for the reason: "Corresponding Angles Postulate (If two parallel lines are cut by a transversal, corresponding angles are congruent)" (or "Transitive Property" if we use \( \angle E \cong \angle CDE \) and \( \angle A \cong \angle CDE \) to get \( \angle A \cong \angle E \), but for \( \angle A \cong \angle CDE \), it's Corresponding Angles Postulate).