Two polygons are similar when there is a sequence of translations, rotations, reflections, and dilations taking one polygon to the other. When the polygons are triangles, we only need to check that both triangles have two corresponding angles to show they are similar.

For example, triangle $ABC$ and triangle $DEF$ both have a 30-degree angle and a 45-degree angle.

We can translate $A$ to $D$ and then rotate around point $D$ so that the two 30-degree angles are aligned, giving this picture:

Then a dilation with center $D$ and appropriate scale factor will move $C'$ to $F$. This dilation also moves $B'$ to $E$, showing that triangles $ABC$ and $DEF$ are similar.