In triangle ABC, BE is the perpendicular from B to AC, and CF is the perpendicular from C to AB…

Step 1. The triangles. Take $\triangle BEC$ and $\triangle CFB$.

1. $\angle BEC = \angle CFB = 90^\circ$.
2. $BC = CB$, the common hypotenuse.
3. $BE = CF$, given.

Step 2. The rule. $\triangle BEC \cong \triangle CFB$ by RHS. By CPCT, $\angle BCE = \angle CBF$, that is $\angle C = \angle B$.

Step 3. The sides. Equal angles have equal sides opposite them, so $AB = AC$.

Step 4. The number. $\angle C = \angle B = 58^\circ$, so $\angle A = 180^\circ - 116^\circ = 64^\circ$.