Similarity

Similarity in geometry definition

Properties of Similarity

Two figures are said to be similar if they have the same shape but different sizes. Therefore, similar shapes have the following properties,

• Corresponding angles are equal.
• Corresponding sides are in the same ratio: this means that all sides in one figure must be multiplied by the same number to give the corresponding sides in the other figure.

To have a deep understanding of the application of similarity in geometry, present the similarity theorems for triangles.

Similarity Formulas

Many people (perhaps, you) get excited when there is a formula to solve problems in a topic. These formulas become the identity of these topics and serve to enhance memory retention. However, the concept of similarity lacks this approach. In clearer terms, there is hardly any formula(s) that is attributed to solving the similarity problems.

Nonetheless, problems with similarity are primarily reliant on the understanding and application of the properties of similarity which was discussed in the previous section. More so and more importantly, the understanding, as well as the application of the theorems which are discussed hereafter are indeed much more than having formulas to memorize.

The Similarity in geometry theorems

There are multiple ways in which we can determine whether or not two triangles are similar, by using one of the four triangle theorems.

Angle-Angle similarity

If two angles in a triangle are equal to two angles in another triangle, then these two triangles are similar.

Triangles ABC and DEF are similar, since

$\angle BCA=\angle DFE$

and

$\angle CAB=\angle FDE.$

Angle-Angle similarity, StudySmarter Originals

Side-Angle-Side similarity

If an angle in one triangle is equal to an angle in another triangle and the sides making up this angle are proportional, then these two triangles are similar.

What is meant by proportionality of sides, is that the two sides on triangle ABC must both be multiplied by the same number to give the sides of triangle DEF.

Side- Angle- Side similarity, StudySmarter Originals

The given sides in the above figure have a common ratio, that is,

and the respective angles formed by these corresponding sides are equal,

$\angle ABC=\angle DEF.$

Side-Side-Side similarity

Two triangles could also be classified as being similar in the event that their sides AC, AB, and BC which correspond to the sides of another triangle DF, DE and FE are indeed proportional.

Side-side-side similarity theorem (www.draw.io)

In the diagram, all of the lines forming triangle DEF are the length of their respective side in triangle ABC multiplied by a constant factor r.

Right Angle - Hypotenuse - Side similarity

This theorem is valid only for right-angled triangles.

Two triangles are similar if the length of the hypotenuse and one other side in one triangle are proportional to the length of the hypotenuse and the other side in another triangle. That is

$\frac{BC}{AC}=\frac{EF}{DF}$

Right angle-hypotenuse-side similarity theorem (www.draw.io)

Symbol for Similarity

The symbol we use to show that two things are similar is ∼ . Suppose triangles ABC and DEF are similar, we could then write

Δ ABC ∼ Δ DEF.

Similarity in polygons

Polygons are plane shapes that have three or more sides. This means that a triangle is also a polygon. The concept of similarity also occurs in other polygons other than triangles.

In fact, the similarity of triangles is a particular case of the similarity of polygons.

However, for similarity to occur between polygons, two conditions need to be met:

1. The corresponding angles of the pair in comparison must be equivalent.

2. The corresponding sides of the pair in comparison must have equivalent proportions.