Chapters
Consider the below image.
Do you notice something? They both look similar but are they similar? That is why we will tell you how to find whether two triangles are similar or not.
Similarity and Congruency
Before we head how to distinguish whether two triangles are similar or not, let's learn about similarity and congruency. The word, "congruent", means that two things are equal in all proportion. In simple words, both objects are identical. In the world of geometry, two triangles are congruent if all the sides and angles are the same. Therefore, if you spot a single difference, both triangles will not be called congruent.
Let's take a look at similarity. The word, "similarity", means both objects have a resemblance. For example, you have two pencils. One of the pencils is sharpened that means the pencil is short in length, however, both have the same point, both are from the same company, having the same colour. One thing is for sure, both triangles are not congruent, but we can call them similar. The reason is simple, everything is the same, just the height of the pencil is different. We can conclude that both pencils are similar to each other.
In the world of geometry, two triangles are similar if they have the same shape but different size. Their sides can be different but if you notice carefully, all the sides of similar triangles have the same ratio. Consider the above image, they are not congruent triangles because all the sides are not equal. However, if you notice, all the sides of triangle number two look similar to triangle number one. This means we need to go deeper and then come up with a conclusion that whether both triangles are similar or not. Let's find out!
How To Find Similarity of Both Triangles
Whether you are fixing anything or doing analysis or optimization, you need tools. Tools are important for every field and mathematics is no exception. We will be also using a tool that will help us to figure out whether two triangles are similar or not. The tool that we will use is called the "comparison tool".
We will be needing the above image again for better clarification. We will be comparing each and every side of the triangle, as well as angles, and then check whether we are looking at some kind of pattern or not. There is no specific arrangement here, you can check the sides first or angles. We will start from the sides of the triangle first but if you want, you can go for the angles first.
Two triangles are similar if:
- Their corresponding sides are proportional, that is to say, they have the same ratio.
- The perimeters of similar triangles have the same ratio.
- The ratio of areas of similar triangles is equal to the square of the ratio.
For the case of angles, you need to check that all angles should be equal. If the sides of triangles are increasing or decreasing at a specific ratio then their angles should be the same.
Problems
1. Calculate the height of a building that casts a shadow of 
meters if at the same time and in the same place a pole of 
in height produces a shadow of 
.
2.The legs of a right triangle measure 
and 
. What is the length of the legs of a similar triangle to this one whose hypotenuse is 
?
Similar Triangle Rules
We know that these rules don't look pleasing, we broke the rules into different examples to make it easier for you to understand and learn.
Rule No.1
Two triangles are similar if two angles are equal.
Rule No.2
Two triangles are similar if the sides are proportional.
Rule No.3
Two triangles are similar if two sides are proportional and the angle between them is equal.
Rule No.4
Two right triangles are similar if they have an equal acute angle.
Rule No.5
Two right triangles are similar if they have two legs that are proportional.
Rule No.6
Two right triangles are similar if the ratio of the hypotenuses and the legs are proportional.
Examples
Determine whether the following triangles are similar:
The triangles are similar because the sides are proportional.
They are similar triangles because they have two equal angles.
They are similar because the two sides are proportional and the angle between them is equal.
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I’m just curious if the area between the polygon and the circumscribed circle has a name.
https://www.superprof.co.uk/resources/academic/maths/geometry/plane/orthocenter-centroid-circumcenter-and-incenter-of-a-triangle.html