Which of the following statements correctly defines similarity for polygons? Option (A) two polygons are said to be similar if their corresponding angles are equal. Option (B) two polygons are said to be similar if their corresponding sides are equal. Option (C) two polygons are said to be similar if their corresponding sides are congruent. Option (D) two polygons are said to be similar if their corresponding angles are congruent and their corresponding sides are in proportion. Or option (E) two polygons are said to be similar if their corresponding angles are complementary and their corresponding sides are equal.
So let’s begin by considering some pairs of polygons which might be said to be similar. So for example, we have two squares which are similar, and here we have two similar triangles. The important thing to remember is that there are two different words, similar and congruent. Congruent polygons are exactly the same shape and size, but similar polygons can be a different size.
So how exactly do we define similar polygons? Well, let’s begin by looking at the two squares. Let’s say that we try to draw another shape, another square, which is similar to the first smaller square. If the definition of similar polygons is that corresponding angles have to be equal, then we could in theory create another shape like this, which would be a rectangle. If it was only that corresponding angles have to be equal, then we could have created this rectangle, but we know that this is not the case. The rectangle cannot be similar to the square. This is what is mentioned in the option given in (A). So this definition is incorrect.
Next, let’s consider the options that are given in options (B) and (C) regarding that corresponding sides have to be equal or congruent. So let’s take this smaller square again and see if we can create a similar shape simply by drawing a shape with congruent sides. And here we have one drawn. But of course this one would be classed as a rhombus, and we know that a rhombus is not similar to a square. And therefore, we can eliminate options (B) and (C).
So let’s look at option (D). This option says that two polygons are similar if their corresponding angles are congruent and their corresponding sides are in proportion. In fact, this is exactly what is demonstrated in the two sets of drawings. Within the squares, all corresponding angles are congruent; they’re all 90 degrees. And the corresponding sides must be in proportion. In these squares, in fact, the larger square has sides which are all double the corresponding side lengths in the smaller square.
Within the pair of triangles, the corresponding angle measures are congruent. And if we label the sides in the smaller triangle as 𝑎, 𝑏, and 𝑐, then each side in the larger triangle is three times the corresponding side length in the smaller triangle. We could say that these side lengths are three 𝑎, three 𝑏, and three 𝑐 length units. It will be this answer given in option (D) which is the correct definition for similar polygons.
But let’s have a look at the final answer option (E). This option tells us that polygons are similar if corresponding angles are complementary and corresponding sides are equal. This sounds like a plausible mathematical definition, but let’s recall what complementary angles are. Complementary angles are angles which sum to 90 degrees. Let’s take an example of this equilateral triangle which has three angles of 60 degrees. If option (E) is a correct definition, then by trying to create a similar triangle, this triangle would need to have three angles of 30 degrees because 30 degrees adds to 60 degrees to give 90 degrees.
But of course, we can’t create a triangle which has three 30-degree angles, since we know that the angle sum in a triangle must be 180 degrees. And even if we tried to apply this rule to something like this square which has four 90-degree angles, if the angles in a similar shape had to be complementary, then we’d be trying to create a square which has four zero-degree angles. Therefore, we can eliminate answer option (E). This leaves us with the answer that two polygons are said to be similar if their corresponding angles are congruent and their corresponding sides are in proportion.