Question

Given: A, B, C, DABCD is a parallelogram, start overline, A, E, end overline, is perpendicular to, start overline, E, C, end overline
AE
⊥
EC
and start overline, D, F, end overline, is perpendicular to, start overline, E, C, end overline, .
DF
⊥
EC
.

Prove: A, E, F, DAEFD is a parallelogram.
In the table below, the first step of the proof is completed for you. To complete each following step, first use the dropdown menu in the "Statement" column to select the format of a statement. Once you select a format, a statement will appear with input boxes and/or dropdowns to complete, and a dropdown menu will appear in the reason column. Once you complete these inputs and dropdowns, navigate back to the first column and press the "try" button. If your statement and reason are valid, a new row will appear below. This process will continue until you have filled in a statement and reason for what you were asked to prove, and then the "try" button will cause the answer area to appear with a submit button. The diagram associated with this problem is located below the proof table, and is available as a tactile printout. There may be a clarifying note below the diagram.
Step Statement Reason
1
A, B, C, DABCD is a parallelogram
start overline, A, E, end overline, is perpendicular to, start overline, E, C, end overline
AE
⊥
EC

start overline, D, F, end overline, is perpendicular to, start overline, E, C, end overline
DF
⊥
EC

Given. What are the remaining steps, statements, and reasons for this question and PLEASE say them correctly!

Answer 1

Here is a clear step-by-step proof (statements with reasons) that AEFD is a parallelogram.

1. Statement: ABCD is a parallelogram; AE ⟂ EC; DF ⟂ EC.
Reason: Given.

2. Statement: AE ∥ DF.
Reason: If two lines are both perpendicular to the same line, they are parallel. (AE ⟂ EC and DF ⟂ EC ⇒ AE ∥ DF.)

3. Statement: E, F, C are collinear (so EF is the same line as EC).
Reason: E and F lie on line EC (by the construction/diagram: EC is the line through E and C and F is the foot of the perpendicular from D to EC).

4. Statement: AD ∥ BC.
Reason: Opposite sides of a parallelogram are parallel (property of parallelogram ABCD).

5. Statement: EF ∥ BC.
Reason: EF lies on line EC and (from the diagram) EC is part of BC (E and F are points on BC); therefore EF is collinear with BC and so EF ∥ BC. (Or: E and F lie on BC, so EF is a segment of BC.)

6. Statement: EF ∥ AD.
Reason: If EF ∥ BC and BC ∥ AD, then EF ∥ AD (transitivity of parallelism).

7. Statement: AEFD is a parallelogram.
Reason: A quadrilateral is a parallelogram if both pairs of its opposite sides are parallel. Here AE ∥ DF (step 2) and EF ∥ AD (step 6), so AEFD is a parallelogram.

This completes the proof.