If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend.
Proof
Same scheme as VI.5: extend one triangle so as to match the second
on the equal-angle pair (I.23), apply VI.4 to deduce the missing
side, then I.4 (SAS) for congruence of the auxiliary triangle with
the second.