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Understanding proportional side lengths is crucial when determining the similarity between polygons. Similar polygons will have sides that correspond to each other in such a way that there is a constant ratio, or scale factor, between them. This means if you multiply the lengths of one polygon's sides by this scale factor, you will get the lengths of the corresponding sides of the other polygon.

For example, consider two triangles with sides 3 cm, 4 cm, and 5 cm, and 6 cm, 8 cm, and 10 cm respectively. They are similar because their side lengths are in the ratio of 1:2; each side of the second triangle is exactly twice the length of the corresponding side of the first triangle. When two polygons are stated to be similar, such as in the exercise about isosceles trapezoids, we must check their respective side lengths to verify this proportionality.

Another essential aspect of similar polygons is having congruent angles. When polygons are similar, not only are their side lengths proportional, but their corresponding angles are also equal in measure. This condition reinforces the fact that similar polygons are basically the same shape, but possibly with different sizes.

To illustrate, let's say we have two quadrilaterals. If one has angles of 90°, 90°, 120°, and 60°, and another has angles of 90°, 90°, 120°, and 60° as well, then their angles are congruent. This congruence, combined with proportional side lengths, will suggest that these quadrilaterals are similar. In our exercise dealing with isosceles trapezoids, we see that the angles themselves may indeed be congruent. However, without side length proportionality, we cannot declare the trapezoids similar.

Focusing on isosceles trapezoids, which are a specific type of polygon, their properties give us clues about their potential similarity with other trapezoids. An isosceles trapezoid is characterized by having at least one pair of parallel sides, referred to as the bases, and non-parallel sides, or legs, that are of equal length. Moreover, the base angles (the angles that are adjacent to each base) are congruent.

Due to these properties, isosceles trapezoids will always have congruent angles by definition. However, for two isosceles trapezoids to be considered similar, as stated in our exercise, their corresponding side lengths must also be proportional. Therefore, while the angle criterion for similarity is automatically met due to the shape's definition, the criterion for side ratio needs to be verified to confirm if two isosceles trapezoids are always, sometimes, or never similar.