Question

given: abcd is a parallelogram, \\(\overline{ae} \perp \overline{ec}\\) and \\(\overline{df} \perp \overline{ec}\\).
prove: \\(\overline{eb} \cong \overline{fc}\\).

step	statement	reason
2	\\(\angle e\\) is a right angle	perpendicular lines form right angles
3	\\(\angle cfd\\) is a right angle	perpendicular lines form right angles
4	\\(\angle e \cong \angle cfd\\)	all right angles are congruent
5	\\(\overline{ab} \cong \overline{dc}\\)	opposite sides of a parallelogram / rhombus / rectangle / square are congruent
6	\\(\overline{ab} \parallel \overline{dc}\\)	opposite sides of a parallelogram / rhombus / rectangle / square are parallel
7	\\(\angle eba \cong \angle c\\)	parallel lines cut by a transversal form congruent alternate interior angles
8	\\(\triangle eba \cong \triangle fcd\\)	aas
9	\\(\overline{eb} \cong \overline{fc}\\)	corresponding parts of congruent triangles are congruent (cpctc)

the proof is complete. you may now press submit. you can still change reasons.
image of the geometric figure with points e, b, f, c and right angles at e and f

Explanation:

The proof is structured to show that \(\triangle EBA \cong \triangle FCD\) using the AAS (Angle - Angle - Side) congruence criterion and then use CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to prove \(\overline{EB} \cong \overline{FC}\).

Step 1: Identify Given Information

We are given that \(ABCD\) is a parallelogram, \(\overline{AE}\perp\overline{EC}\) and \(\overline{DF}\perp\overline{EC}\). These are the starting points of our proof.

Step 2: Prove Right Angles

Since \(\overline{AE}\perp\overline{EC}\) and \(\overline{DF}\perp\overline{EC}\), by the definition of perpendicular lines (perpendicular lines form right angles), \(\angle E\) and \(\angle CFD\) are right angles. And all right angles are congruent, so \(\angle E\cong\angle CFD\).

Step 3: Use Parallelogram Properties

In a parallelogram \(ABCD\), opposite sides are congruent and parallel. So \(\overline{AB}\cong\overline{DC}\) and \(\overline{AB}\parallel\overline{DC}\). When \(\overline{AB}\) and \(\overline{DC}\) are cut by the transversal \(\overline{BC}\) (or the line containing \(\angle EBA\) and \(\angle C\)), by the alternate - interior angles theorem (parallel lines cut by a transversal form congruent alternate interior angles), \(\angle EBA\cong\angle C\).

Step 4: Prove Triangle Congruence

We have \(\angle E\cong\angle CFD\) (right angles), \(\angle EBA\cong\angle C\) (alternate interior angles) and \(\overline{AB}\cong\overline{DC}\) (opposite sides of a parallelogram). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle EBA\cong\triangle FCD\).

Step 5: Use CPCTC

Since \(\triangle EBA\cong\triangle FCD\), by the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, the corresponding sides \(\overline{EB}\) and \(\overline{FC}\) are congruent.

Answer:

The proof shows that \(\triangle EBA\cong\triangle FCD\) by AAS and then by CPCTC, \(\overline{EB}\cong\overline{FC}\) is proven. So \(\overline{EB}\cong\overline{FC}\) is a valid conclusion based on the given information and the properties of parallelograms, right angles, and triangle congruence.