Triangles with One Equal Angle and Two Other Sides Proportional are Similar

Theorem
Let two triangles be such that one of the angles of one triangle equals one of the angles of the other.

Let two corresponding sides which are adjacent to one of the other angles, be proportional.

Let the third angle in both triangles be either both acute or both not acute.

Then all of the corresponding angles of these triangles are equal.

Thus, by definition, such triangles are similar.


 * ''If two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles either both less or both not less than a right angle, the triangles will be equiangular and will have those angles equal, the sides about which are proportional.

Proof
Let $\triangle ABC, \triangle DEF$ be triangles such that:
 * $\angle BAC = \angle EDF$
 * two sides adjacent to $\angle ABC$ and $\angle DEF$ proportional, so that $AB : BC = DE : EF$
 * $\angle ACB$ and $\angle DFE$ either both acute or both not acute.