Similarity

In this lesson, you will learn that there are triangles and other polygons that have the same shape but do not necessarily have the same size. The illustrative example below will give you an idea on how we can say that the given figures are similar.

If you will observe, trapezoid ABCD and trapezoid EFGH have the same shape. When you pair the corresponding vertices, the angles coincide. It shows that their corresponding angles are congruent: ∠A ≅∠E ; ∠B ≅ ∠F ; ∠C ≅ ∠G ; and ∠D ≅ ∠H.

Another thing is that the ratios of the measure of the lengths of their corresponding sides are equal.

Based on the illustrative example, two polygons are similar (the symbol is ∼) if their vertices can be paired so that corresponding angles are congruent and the lengths of their corresponding sides are proportional.

To indicate that trapezoid ABCD is similar to trapezoid EFGH, you can write ABCD ∼ EFGH. If you use this notation, write the corresponding vertices on the same order.

Example:

Triangle Similarity Theorems

In this lesson, we are only going to focus on the similarity of two triangles. We will apply our prior knowledge on the definition of similar polygons to understand the postulates and theorems in proving the similarity of triangles.

To prove the similarity of two triangles using the definition of similarity, we must establish that the three corresponding angles are congruent and that the three ratios of the lengths of corresponding sides are equal.

If the three corresponding angles of two triangles are congruent, then we can conclude that the triangles are similar. We call this as AAA Similarity Theorem.

Here are the other Triangle Similarity Theorems.

1.1 SAS Similarity Theorem

If an angle of one triangle is congruent to an angle of another triangle and if the lengths of the sides including these angles are proportional, then the triangles are similar.

1.2 SSS Similarity Theorem

If the three corresponding sides of two triangles are proportional, then the two triangles are similar.

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