Similar Triangles

When it comes to geometry, ‘Triangle’ is one of the most elementary sections for students. Students get more than 50+ theorems on triangles and each theorem present properties. These properties include angle sum property of triangle, Pythagoras theorem, congruence theories and so on. In these articles you will learn more about similar triangles and their properties along with images.

That means similar shapes when magnified or de-magnified overlay each other. This property of similar shapes is referred to as "Similarity".

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What are Similar Triangles?

A: Similar Triangles Definition

Similar triangles and its problems are very common in geometry and math students generally get confused with the variety of triangles. In other words, Similar triangles are those triangles that have the same shape but their sizes vary. It implies that both triangles have corresponding sides in proportion to each other and corresponding angles equal to each other. In general, similar triangles are different from congruent triangles. Two objects can be said similar if they have the same shape but might vary in size.

Indeed, there are different methods and theories by which you can find if two triangles are similar or not.

Before you move on to you similar triangles example, here is the definition of “equilateral triangle”.

An equilateral triangle has all three sides equal to each other. Due to this all the internal angles are of equal degrees, i.e. each of the angles is 60°.

Here is the image:

Here you can see, AB= BC and ∠ABC and ∠ACB are also equal.

B: Similar Triangles Examples

Similar triangles are triangles for which the corresponding angle pairs are equal. That implies equiangular triangles are similar.

Hence, all equilateral triangles are examples of similar triangles. The following image shows similar triangles, but we must notice that their sizes are different.

Similar Triangles Formulas

You have already learned about the condition of being similar triangles. There are two conditions using which we can verify if the given set of triangles is similar or not.

These conditions highlights that two triangles can be said similar if either their corresponding angles are equal or congruent or if their corresponding sides are in proportion.

Therefore, two triangles â–³ABC and â–³PQR can be proved similar that is â–³ABC ∼ â–³PQR using either condition among the following set of similar triangles formulas,

Formula for Similar Triangles in Geometry:

∠A = ∠P, ∠B = ∠Q and ∠C = ∠R

AB/PQ = BC/QR = AC/PR

3: Similar Triangles Theorems:

You easily can find out or prove whether two triangles are similar or not using the similarity theorems. These theorems are also useful while you will attempt geometrical problems related to square, rectangle, right-angle triangle, polygon and so on. Generally, mathematicians use these similarity criteria when they do not have the measure of all the sides of the triangle or measure of all the angles of the triangle.

These similar triangle theorems are useful to you to find out whether two triangles are similar or not. There are three major types of similarity rules, as given below,

A: AA (or AAA) or Angle-Angle Similarity Criterion

AA or AAA similarity theorem states that if any two angles in a triangle are respectively equal to any two angles of another triangle, then they must be similar triangles.

This rule is easily useful when you only know the measure of the two angles and have no idea about the length of the sides of the triangle.

In the image given below, if it is known that ∠B = ∠Q, and ∠C = ∠R.

And we can say that by the AA similarity criterion, â–³ABC and â–³PQR are similar or â–³ABC ∼ â–³PQR.
⇒AB/PQ = BC/QR = AC/PR and ∠A = ∠P.

B: SAS or Side-Angle-Side Similarity Criterion

According to the SAS or side-Angle-Side similarity theorem, if any two sides of the first triangle â–³ABC are in exact proportion to the two sides of the second triangle â–³PQR along with the angle formed by these two sides of the individual triangles are equal, and then they must be similar triangles. It is the rules generally applied when we only know the measure of two sides and the angle formed between those two sides in both the triangles respectively.

In the image given below, if it is known that AB/PQ = AC/PR, and ∠A = ∠P

And we can say that by the SAS similarity criterion, â–³ABC and â–³PQR are similar or â–³ABC ∼ â–³PQR.

C: SSS or Side-Side-Side Similarity Criterion

According to the SSS or Side-Side-Side similarity theorem, two triangles will the similar to each other if the corresponding ratio of all the sides of the two triangles is equal.

It is commonly used when you only have the measure of the sides of the triangle and have less information about the angles of the triangle.

In the image given below, if it is known that PQ/ED = PR/EF = QR/DF

And here, you can say that by the SSS similarity criterion, â–³PQR and â–³EDF are similar or â–³PQR ∼ â–³EDF.

Comparison between similar triangles and congruent triangles

Similar triangles are the triangles that have the different size, but their shapes are similar. Here are the properties of similar triangle:

Both triangles have the same shape but sizes are different

The ratio of corresponding sides is the same

Each pair of corresponding angles is equal

Whereas, we will say when two triangles are said to be congruent if the three sides and the three angles of both the triangles are equal is called congruent triangle. Here,

Their sides have the same length and angles have the same measure.
Thus, two triangles can be overlaid side to side and angle to angle.

How to identify similar triangles?

According to similar triangle theorems, there are three ways to identify similar triangles: And indeed it varies what information are there in your question. If you are given conditions like two sides are given and one angle measurement is also given, then you can use AA theorem.

If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are alike. When the three angle pairs are all equivalent, the three pairs of sides must also be in proportion.

SSS (Side-Side-Side)

Another way to prove triangles are similar is by SSS, side-side-side. If the measures of corresponding sides are known, then their proportionality can be calculated. If all three pairs are in proportion, then the triangles are similar.

SAS (Side-Angle-Side)

If two pairs of corresponding sides are in proportion, and the included angle of each pair is equal, then the two triangles they form are similar. Any time two sides of a triangle and their included angle are fixed, then all three vertices of that triangle are fixed.

You can apply any of these three basic similar triangle theorem to prove your point.