The leghts of AB is 9 because both of the triangles are symmetrical. But also, this is a Equilateral triangle, which means that both of the sides are the same lengths (AB=AC)
In the given two column proof two sides and included angles are equal.
Triangles are congruent if any pair of corresponding sides and their included angles are equal in both triangles .In the figure sides BD≅BD , AB≅ BC and <ABD ≅,BDD Therefore triangle ABD and triangle CBD are congruent by SAS property of congruence.
Since BD is similar to GI, the ratio of the sides of the triangles is 27:3=9:1. This means that BC is 9 times larger than BH, or 4*9=36 units long. Hope this helps!