Summary 3, theory

Apart from the triangle proportionality theorem, its corollary and the definition of similar triangles; you should also know three cases yielding similar triangles:

Side-Angle-Side If two sides in a triangle are in the same ratio to two corresponding sides of another triangle, and if the included angles in both triangles are the same, then the triangles are similar.

Proof

Place the point G such that AG=DE.
Draw GH such that GH is parallel to BC.

Now ΔAGH∼ΔABC and ΔDEF∼ΔABC.

Show that AH=DF!

Then the congruence case Side-Angle-Side yields ΔAGH≅ΔDEF and the proof is done.

The proofs of the other two cases are similar.

Side-Side-Side If three pairs of corresponding sides are in the same ratio then the triangles are similar.

Angle-Angle-Angle If the angles in a triangle are congruent (equal) to the corresponding angles of another triangle then the triangles are similar.

Also note that it suffices that two angles are equal (why?).