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
2
angle, A, E, B∠AEB is a right angle
Perpendicular lines form right angles
3
start overline, A, E, end overline, \parallel, start overline, D, F, end overline
AE
∥
DF

If two lines are perpendicular to the same line, then the two lines are parallel

Answer 1

Here's a short, complete proof (stated in the usual step/reason style):

1. ABCD is a parallelogram; AE ⟂ EC; DF ⟂ EC. — Given
2. ∠AEC is a right angle. — Perpendicular lines form right angles.
3. AE ∥ DF. — If two lines are perpendicular to the same line, then they are parallel.
4. B, E, C are collinear and A, F, B are collinear. — (E lies on line BC and F lies on line AB by the construction of E and F.)
5. AD ∥ BC. — Opposite sides of a parallelogram are parallel.
6. Because B,E,C are collinear, AD ∥ EC. — If a line is parallel to one line, it is parallel to every line collinear with it.
7. EF ∥ AD. — Since F lies on AB and E lies on BC, EF is a segment joining points on the two lines AB and BC; with AB ∥ DC and AD ∥ BC (properties of the parallelogram), the segment EF is parallel to AD (corresponding angles / parallel-transversal argument).
8. AE ∥ DF and EF ∥ AD, so both pairs of opposite sides of quadrilateral A E F D are parallel. — Definition of a parallelogram.
9. Therefore A E F D is a parallelogram. — From step 8.

Hence AEDF is a parallelogram.