Similarity of Triangles and Polygons

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Similarity of triangles and polygons

Similar triangles are triangles whose three angles are equal respectively and also the ratio between each pair of corresponding sides is equal. Two similar triangles are actually larger or smaller versions each other.

The ratio of similarity is the ratio between two corresponding sides in two similar triangles.

To prove similarities between triangles, we will use the following theorems:

  • Angle-Angle (A.A): If two angles are equal respectively between two triangles, then the triangles are similar.
  • Side-Angle-Side (S.A.S): If the ratio of two pairs of sides is equal, and also the angles between them are equal to each other, then the triangles are similar.
  • Side-Side-Side (S.S.S.): If for two triangles, the ratio of the three sides in one triangle to the three pairs in the other triangle is equal (similarity ratio), then the triangles are similar.

For similarity of polygons we will define it this way: if for two polygons all angles are equal and there is a constant ratio between two corresponding sides, then the polygons are similar.

Intuitively, just like similar triangles, also two similar polygons are actually an enlargement or reduction of each other.

Image 1 similar triangles

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If it is known that both triangles are equilateral, are they therefore similar?

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Similar triangles

Definition: Similar triangles are triangles whose three angles are equal respectively and also the ratio of each pair of corresponding sides is equal.
Two similar triangles are actually an enlargement or reduction of each other.

To understand this, let's look at the following example:


Example 1

Given the two triangles in the drawing

ΔABCΔ ABC
ΔDEFΔ DEF

similar triangles

Given that the triangle ΔABCΔ ABC and the triangle ΔDEFΔ DEF are similar triangles.
We will mark this with the sign   ~

It looks like this:
ΔABCΔ ABC ~ It is important to write the correct order of the vertices, similar to the superposition of triangles. ΔDEFΔ DEF

From here we can conclude that the three angles are equal respectively, ie:

A=D∢A=∢D
B=E∢B=∢E
C=F∢C=∢F

And we can conclude that the ratio between each pair of corresponding sides is equal. That is:

ABDE=BCEF=CAFD\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}

This ratio of sides is called the ratio of similarities. It is important to note that two overlapping triangles are also similar triangles when the ratio of similarity is 1 1 .


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What is the similarity ratio?

The similarity ratio is the ratio between two corresponding sides in two similar triangles.

Example 2

Since the two triangles ΔABCΔ ABC and ΔDEFΔ DEF are similar triangles, that is:

ΔABCΔ ABC ~ ΔDEFΔ DEF

It is also given
AB=8AB = 8
BC=12BC = 12
CA=6CA = 6
In addition:
DE=4DE = 4
EF=6EF = 6
FD=3FD = 3
All the data are marked in the drawing.

Calculate the similarity ratio between the two triangles.

Calculate the similarity ratio of the two triangles

Pay attention that we do not know the size of the angles, but we do not need it to calculate the similarity ratio, since it is stated that they are similar triangles, so their respective angles measure the same. We can calculate the similarity ratio by the ratio between each pair of corresponding sides:

ABDE=84=2\frac{AB}{DE}=\frac{8}{4}=2
BCEF=126=2 \frac{BC}{EF}=\frac{12}{6}=2
CAFD=63=2 \frac{CA}{FD}=\frac{6}{3}=2

That is, we have seen that the similarity ratio between the triangle ΔABCΔ ABC For the triangle ΔDEFΔ DEF Is 1:2 1:2 .
QED

Pay attention to the similarity ratio between the lengths of the sides of the triangle ΔDEFΔ DEF For the triangle ΔABCΔ ABC It is 2:1 2:1
Intuitively, the length of each side in a triangle ΔABCΔ ABC is 2 2 times longer than each side in a triangle ΔDEFΔ DEF Respectively.


To prove the similarity between triangles we will use one of the following three theorems:
- Angle-Angle (A.A): If two angles are equal in correspondence between two triangles, then the triangles are similar.
- Side-Angle-Side (S.A.S): If the ratio between two pairs of sides is equal, and also the angles included between them are equal to each other, then the triangles are similar.
- Side-side-side (S.S.S.): If for two triangles, the ratio between the three sides in one triangle to the three pairs in the other triangle is equal (similarity ratio) then the triangles are similar.


Exercise with example - (How to calculate the length of the side)

Given two triangles in the drawing below
ΔABCΔ ABC
ΔDEFΔ DEF
are similar triangles, i.e.
ΔABCΔ ABC ~ ΔDEFΔ DEF
In addition given:
AB=5AB = 5
DE=2.5DE = 2.5
FD=1FD = 1
A=D∢A=∢D
B=E∢B=∢E
C=F∢C=∢F

All data are labeled in the drawing.

All data are marked on the drawing.

Question: what is the length of the side AC AC ?

Solution:
The two triangles are similar, so we will calculate the similarity ratio and use it to solve the assignment. Remember that the ratio between two sides in similar triangles is equal and therefore:

ABDE=52.5=21\frac{AB}{DE}=\frac{5}{2.5}=\frac{2}{1}

That is, the similarity ratio is 2:1 2:1 , and each side of the triangle ABC \triangle ABC is twice as large as any corresponding side in the triangle DEF \triangle DEF .
Now we can calculate the length of side AC. According to the similarity ratio:

ACDF=2 \frac{AC}{DF}=2

We replace and obtain:

AC1=2 \frac{AC}{1}=2

That is, we obtained:

AC=2AC=2

QED


Do you know what the answer is?

Similar polygons

Definition: If for two polygons all angles are equal and there is a constant ratio between two corresponding sides, then the polygons are similar.
Intuitively, as in similar triangles, two similar polygons are actually larger and smaller versions of each other.


Example 3 - Similar polygons

These two squares are similar squares:

These two squares are similar squares

Any two corresponding angles are equal since all angles are equal. The ratio of the two corresponding sides, i.e., the similarity ratio, is 2/12/1 or, in other words, each of the sides is twice as large for the large square as for the small square.


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Example 4 - Similar polygons

Two pentagons in the drawing are similar, which means that any pair of corresponding angles are equal. When the similarity ratio is

FGAB=32=1.51 \frac{FG}{AB}=\frac{3}{2}=\frac{1.5}{1}

Two pentagons in the drawing are alike

That is, for each pair of corresponding sides, the length of the pentagon FGHIJFGHIJ is 1.51.5 times greater than that of the pentagon ABCDE ABCDE .


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Exercises on similarity of triangles and polygons

Exercise 1

Task
Given:

D=60° ∢D=60°

E=70° ∢E=70°

AC=12 AC=12

AE=24 AE=24

AB=15 AB=15

AD=30 AD=30

Are the triangles similar?

Exercise 1 Given ∢D=60 ,∢E=70

Solution

ABAD=ACAE \frac{AB}{AD}=\frac{AC}{AE}

We replace using the data

1530=1224=12 \frac{15}{30}=\frac{12}{24}=\frac{1}{2}

A \sphericalangle A common

Answer

Yes, according to S.A.S S.A.S


Do you think you will be able to solve it?

Exercise 2

Task

Given that ABCBCD ABC∼BCD

Choose the correct answer

Exercise 2 Given that ABD∼BCD

Solution

Given that ABCBCD ABC\sim BCD

B1=B2 \sphericalangle B_1=\sphericalangle B_2

BC BC common

Therefore

BCBC=ABBD=1 \frac{BC}{BC}=\frac{AB}{BD}=1

AB=BD AB=BD

Answer

AB=BD AB=BD


Exercise 3

Question

Is it possible to say that the two triangles are similar?

It is possible to say that the two triangles are similar.

Solution

There is no data about the sides AB AB and DE DE

and there is no data on the rest of the angles.

Answer

No, it is not impossible to know


Test your knowledge

Exercise 4

Task

Given that the two triangles are isosceles

and the angles of the head A=F ∢A=∢F

Are they ABCFDE ABC∼FDE ?

Exercise 4 - Given that the two triangles are isosceles

Solution

If F=A \sphericalangle F=\sphericalangle A

and the two triangles are isosceles then so are the top angles.

B=C=E=D \sphericalangle B=\sphericalangle C=\sphericalangle E=\sphericalangle D

Answer

Yes, according to A.A A.A


Exercise 5

Question

If it is known that the two triangles are equilateral, are they similar?

two triangles are equilateral

Solution

Yes, according to A.A A.A the two triangles are similar, because since they are equilateral triangles, then their three angles measure the same.

Answer

Yes


Do you know what the answer is?

Review questions

What is similarity of triangles?

Two triangles are similar if their three respective angles have the same measure and the ratio between the pairs of their respective sides is the same.


What are the three criteria for similarity of triangles?

There are three criteria to determine if two triangles are similar or not, which are the following:

  • Side-Side-Side (SSS): If the ratio of their three pairs of corresponding sides is the same then two triangles are similar.
  • Side-Angle-Side (SAS): Two triangles are similar if the ratio of two pairs of corresponding sides is the same and the angle between these two pairs is the same, then they are similar triangles.
  • Angle-Angle (AA): For two triangles to be similar by this criterion, two of their respective angles must measure the same and therefore the third angle must also have the same measure as the angle corresponding to that angle. That is, their three corresponding angles measure the same.

How to find the similarity ratio?

In order to get the ratio of similarity we must calculate the ratio or division of each of their respective pairs of sides, and this relationship is the same for each of the pairs of sides when two triangles are similar.


When are two triangles similar by the SSS criterion?

Two triangles are similar when they have the same shape but their sides do not necessarily have to measure the same, they are similar by the SSS criterion as long as their corresponding sides are proportional, i.e. the ratio between their sides is the same.


What is the similarity of polygons?

In the same way that the similarity of triangles, two polygons are similar when they have the same shape, they do not necessarily have to have the same measure in their sides, that is, their corresponding angles are equal and the ratio of their corresponding sides is the same for all of them.


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