Write a paragraph proof that triangles BHD and FHC are congruent.
Given:
Triangle ABC is isosceles with AB = AC.
AD is the perpendicular bisector of BC.
E is the midpoint of BC.
Triangle AED is congruent to Triangle AEF by SAS.
Proof:
Since Triangle AED is congruent to Triangle AEF by SAS, we have AD = AF (by corresponding parts of congruent triangles).
We know that BD is congruent to CD since Triangle ABC is isosceles, so BE = EC).
Since Triangle BDE is congruent to Triangle CDE by SSS, we have angle BDE = angle CDE.
Subsequently, angle BDC is congruent to angle BED because they are vertical angles.
However, angle BDC is supplementary to angle CDF since triangle BDC is isosceles, and angle BED is supplementary to angle CFE since triangle BEC is isosceles
Thus, angle CDF is congruent to angle CFE.
Using vertical angles again, we know that angle FDH is congruent to angle FHD.
Therefore, triangle BHD is congruent to triangle FHC by angle-side-angle (ASA) congruence.