Proving if a set of points in a hexagon is a quadrilateral

[Ypara]

Given: ABCDEF is a regular hexagon
Prove that ACDF is a rectangle

My first answers I got were:

1)ABCDEF is a regular hexagon 1R) Given S2) Sides EF , FA, AB, CB, DC, ED are congruent R2) All sides of a regular hexagon are congruent S3) Angles FED congruent angle ABC R3) All interior angles of a regular hexagon are congruent S4)Triangle FED congruent triangle ABC R4) SAS S5) Segment FD congruent Segment AC S5) CPCTC S6) Angle EFD congruent angle BCA S6) CPCTC S7) segment FD congruent segment AC S7) AIATC S8) Sides FA and DC are parallel S8) Opposite sides of a hexagon are parallel S9) ACDF is a rectangle S9) Opposite sides are parallel and congruent

On my assessment, I got 1/3 points on this proof.

 

[Dr.Peterson]

S6 doesn't technically follow from [MATH]\triangle FED \cong \triangle ABC[/MATH], but rather [MATH]\triangle DEF \cong \triangle ABC[/MATH].

S7 surely was intended to be different from S5, probably "segment FD parallel to segment AC". Moreover, though I'm not sure what your "AIATC" theorem says exactly, you don't have a single transversal here, so probably more is needed.

S9 proves only that ACDF is a parallelogram. You've shown nothing about right angles, which is the key piece you are missing.

 

[Ypara]

Ok thank you so much! The AIATC is an acronym my teacher uses for alternate interior angles theorem converse

 

[pka]

Given: ABCDEF is a regular hexagon
Prove that ACDF is a rectangle
View attachment 12966

As pointed out you need to show that \(\displaystyle \angle FDC\) is a right angle.
Can you show that \(\displaystyle \Delta FED\) is Isosceles with summit angle of \(\displaystyle 120^o~?\)
If you can the what are its base angles?
Then \(\displaystyle \angle FDC=~?\).

 

[JeffM]

Dr. P as usual is correct that you need to show that angle CAF is a right angle. It is trivial to show that AF = CD. By showing that triangles ABC and DEC are congruent, you can show that AC = FD.

What you neglected to consider is that those triangles are isoceles.