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Two polygons are similar iff ...

We say two figures are similar if they have the same shape, but not necessarily the same size.
If they also have the same size, we say they are congruent.

Which of these statements are true?
  • If two figures are similar, then they are also congruent.
  • If two figures are congruent, then they are also similar.

There are two conditions (tests) for two polygons to be similar:

  1. all the corresponding angles must be equal, and
  2. all the corresponding sides must be proportional.

So the definition is: Two polygons are similar iff they are equiangular and their corresponding sides are proportional. Let's look at the importance of satisfying both conditions for polygons.

Are equiangular polygons similar?
This applet provides a quick and definite answer.

The green quadrilateral has its vertices on lines parallel to the sides of the blue quadrilateral. So the two quadrilaterals are equiangular.

But do they have the same shape?
Drag the red points and judge visually ...! 		Sorry, this page requires a Java-compatible web browser.

Do they have the same shape?
Open the Similar Quadrilaterals applet:

In the applet, the sides of the smaller quadrilaterals are parallel to the sides of the larger quadrilaterals, so the quadrilaterals are equiangular in both cases. Interact with the applet and explain why:

  • The sides of AB¢C¢D¢ and ABCD are nearly never proportional, so they are nearly never similar, i.e. they do not have the same shape (find at least one case for which they are similar).
  • The sides of PQ¢R¢S¢ and PQRS are always proportional, so they are always similar, i.e. they have the same shape.

Two counter examples
Open the Similar or not? applet:

The applet shows two simple examples to show conclusively that for polygons to be similar, they must be both equiangular and proportional.

  • In the applet, drag points P and B … Do you agree that:
    In the left figure, rectangles ABCD and APQD are always equiangular, but their sides are nearly never proportional, so they are not similar (find at least two positions for which the rectangles are similar).
  • In the right figure, the sides of rectangle ABCD and parallelogram APQD are always proportional, but they are nearly never equiangular, so they are not similar (find two positions for which they are similar).