75 points for answwer
given that in isosceles △abc, base bc, be⊥ac, and cf⊥ab, which of the following proves that be≅cf?

.  isosc. △abc, base  bc (given)
2.  ab≅bc (def. of isosc. △)
3.  be⊥ac,  cf⊥ab (given)
4.  ∠bec and ∠cfb are supplementary angles  (def. of ⊥)
5.  ac≅bc (supp. ∠s  ≅  thm.)6.  ∠a≅∠b (sym. prop. of≅)7.  △abe≅△acf (sas steps 1, 6, 5)8.  be≅cf (cpctc)

1.  isosc. △abc, base  bc (given)
2.  ab≅ac (def. of isosc. △)
3.  be⊥ac,  cf⊥ab (given)
4.  m∠bec=m∠cfb=90∘ (def. of ⊥)
5.  ∠bec≅∠cfb (rt. ∠s  ≅ thm.)
6.  ∠a≅∠b (sym. prop. of≅)
7.  △abe≅△acf (aas steps 5, 6, 1)
8.  be≅cf (cpctc)

1.  isosc. △abc, base  bc (given)
2.  ab≅ac (def. of isosc. △)
3.  be⊥ac,  cf⊥ab (given)
4.  m∠bea=m∠cfa=90∘ (def. of ⊥)
5.  ∠bea≅∠cfa (rt. ∠s  ≅ thm.)
6.  ∠a≅∠a (reflex. prop. of≅)7.  △abe≅△acf (aas steps 5, 6, 1)8.  be≅cf (cpctc)

1.  isosc. △abc, base  bc (given)
2.  ab≅bc (def. of isosc. △)
3.  be⊥ac,  cf⊥ab (given)
4.  ∠bea and ∠cfa are supplementary angles  (def. of ⊥)
5.  ab≅bc (supp. ∠s  ≅  thm.)
6.  ∠a≅∠a (reflex. prop. of≅)
7.  △abe≅△acf (sas steps 1, 6, 5)8.  be≅cf (cpctc)