This statement is false.
Here is counter example. Let a parallelograms ABCD and EFGH have angle A and angle E congruent (say 45 degrees). Let the side ratio of the AB:BC= 2:1 and EF:FG=2:3. The linear ratio of the sides is different, therefore the two parallelograms have corresponding congruent angles, and are not similar.
We can have similar construction using trapezoids. If two trapezoids have corresponding angles congruent, and they may not be similar. Here how we can construct a pair of these: we can start with a trapezoid. We can fix one of the legs. If we slide outward the other leg, on any distance, we created a trapezoid with the same angles, but different ratio of the bases.
In a matter of fact, we can do similar construction to many other polygons, constructing 2 polygons with congruent angles, but different sides ratio.
The only figure for which this is not possible is a triangle. If we construct two triangles with the same angles, they must have the same ratio of the sides.
