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Similarity

From Earth, the sun and the moon seem to be exactly the same size. Although the sun dwarfs the moon, this appears to happen because both spheres are similar figures!

This article will cover the definition of similarity in geometry and its applications.

Similarity in geometry definition

Similarity can be defined as an attribute exhibited by two or more figures when their shapes are the same.

An individual is up for a red-night game with his friends requiring them to blindfold each other and make a selection for a similar pair among 4 pastries; a doughnut, a burger bread, sliced bread, and a samosa. Determine the appropriate selection.

Solution:

A samosa is triangular in shape; a slice of bread is rectangular in shape; a burger bread is circular in shape, and a doughnut is circular in shape.

Hence the appropriate similar pair is the burger bread and doughnut.

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,

$\frac{\overline{)DE}}{\overline{)AB}}=\frac{\overline{)EF}}{\overline{)BC}}$

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)

When we use a side in a similarity theorem (for example in the SAS theorem), we do not mean that the sides are equal, but that the ratio between the triangle's sides is constant.

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.

Triangle ABC has sides AB = 6 cm, AC = 4 cm, and BC = 10 cm. Triangle DEF has sides DE=3 cm, DF = 2 cm, and EF = 5 cm. Prove that these triangles are similar.

Solution:

Since we are only given sides, we want to use the SSS similarity theorem.

For us to be able to apply this theorem, we need to find a common ratio between the sides of triangle ABC and triangle DEF.

The ratio between the sides AB and DE is

$\frac{AB}{DE}=\frac{6}{3}=2:1$

The ratio between sides AC and sides DF is

$\frac{AC}{DF}=\frac{4}{2}=2:1$

The ratio between sides BC and EF is

$\frac{BC}{EF}=\frac{10}{5}=2:1$

Since the ratio between the sides of triangle ABC to its respective sides on triangle DEF is constant, we can say that $△ABC~△DEF$

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.

Prove that these two rectangles are similar.

An illustration on polygons with similarity, StudySmarter Original

Solution:

Both rectangles have all their internal angles as right angles. This means that the first criterion which says that all corresponding angles must be equal has been met.

Next, we need to confirm that the ratio of their corresponding sides is equal.

The ratio of both widths is

$\frac{9}{3}=3:1$

and the ratio of both lengths is

$\frac{15}{5}=3:1$

Similarity in geometry examples

To further understand the concept of similarity, here are a few examples.

Determine similarity among the following pairs,

(a)

Using the angle-angle rule, StudySmarter Originals

(b)

Using the side-angle-side rule, StudySmarter Originals

(c)

Using the side-side-side rule, StudySmarter Originals

(d)

Using right angle-hypotenuse-side rule, StudySmarter Originals

Solution:

(a) Using the angle-angle rule, we can tell that both triangles in figure (a) are similar because knowing that the sum of angles in a triangle is 180º, hence the third angle in the first triangle is

$180°-\left(63°+73°\right)=180°-136°=44°$

This confirms that both triangles are similar since all corresponding angles are equal.

(b) The pair in figure (b) are not similar, although the ratio between corresponding sides is equal to 2:1, the corresponding angles between them are different and as such using the side-angle-side rule we can confirm that the triangular pair are not similar.

(c) The pair in figure (c) are not similar because the ratio of the two sides is 2:1 while the ratio of the third side is 5:3. By considering the side-side-side rule, the ratio of all corresponding sides must be equivalent, hence this pair of triangles are not similar.

(d) The pair in figure (d) is similar because they are both right triangles and the ratio of both corresponding hypotenuses and opposite sides is 1:4. This is in compliance with the right angle-hypotenuse-side rule of similarity.

Similarity - Key takeaways

• Figures are similar if they have the same shape.
• There are four similarity theorems for triangles: Angle-angle, side-angle-side, side-side-side, and right angle-hypotenuse-side.
• If two triangles are similar, their respective sides are of proportionate length.
• For two similar triangles ABC and DEF, we write Δ ABC ∼ Δ DEF.

Two figures are similar if they have the same shape.

Two major rules are considered for similarity; the shape between objects in comparison be the same and the proportion of their corresponding sides must be equivalent.

Similar shapes have equal corresponding angles and proportional sides.

There is no exact formula for calculating similarity in geometry, however, the similarity is determined based on compliance with some rules or theorems.

An example of similarity in geometry is two triangles that have equivalent corresponding angles and the proportion of their corresponding sides is the same for all three sides.

Final Similarity Quiz

Question

Given that , , and , state the similarity theorem that proves they are similar.

RHS

Show question

Question

Are the following triangles similar? If so, which similarity theorem proves this?

Triangles (www.draw.io)

Yes-SSS

Show question

Question

Are the following triangles similar? If so, which similarity theorem proves this?

Triagles (www.draw.io)

No

Show question

Question

Are the following triangles similar? If so, what similarity theorem supports this?

Triangles (www.draw.io)

No

Show question

Question

These two triangles are similar. Find the length of line x.

Similar triangles (www.draw.io)

x=13.5

Show question

Question

These two triangles are similar. Find the length of line x.

Similar triangles (www.draw.io)

x=8

Show question

Question

These two triangles are similar. Find the length of line x.

Similar triangles (www.draw.io)

x=42

Show question

Question

Define similarity.

Similarity in geometry is when two shapes can be the same after being flipped, turned, or stretched

Show question

Question

These two triangles are similar. Find the length of line x.

Similar triangles (www.draw.io)

x=18

Show question

Question

These two triangles are similar. Find the area of the larger triangle.

Similar triangles (www.draw.io)

A=60 units squared

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Question

These two triangles are similar. Find the area of the larger triangle.

Similar triangles (www.draw.io)

A=300 units squared

Show question

Question

These two triangles are similar . Find the area of the smaller triangle.

Similar triangles (www.draw.io)

A=1.5 units squared

Show question

Question

What are the four triangle similarity theorems?

RHS, SSS, SAS, AA

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Question

When two polygons have equivalent sides, then, what other factor would determine that they are similar?

Their corresponding sides but be equivalent in proportion.

Show question

Question

What properties certify that two figures are similar?

Corresponding angles must be equal and corresponding sides are in the same ratio.

Show question

Question

All of the following are theorems which determine similarity in triangles except?

Side-Side

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Question

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 theorem proves this?

Side-Angle-Side

Show question

Question

What symbol is used to express similarity?

Show question

Question

What does SAS mean?

Side-Angle-Side

Show question

Question

What does SSS mean?

Sise-Side-Side

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Question

Other polygons except triangle cannot show similarity?

FALSE

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Question

with two internal angles  and  is compared with  with internal angles  and to be similar. Are both triangles similar?

YES

Show question

Question

with two internal angles  and  is compared with  with internal angles  and  to be similar. Are both triangles similar?

NO

Show question

Question

has AB = 5cm and BC 12cm is said to be similar with with PQ=15cm and QR=36cm. Is this statement correct?

YES

Show question

Question

has AB = 5cm and BC 12cm is said to be similar with  with PQ=15cm and QR=36m

NO

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Question

What does RHS mean in similarity?

Right Angle-Hypotenuse-Side

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Question

How many similarity theorems are used in proving that triangles are similar?

4

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Question

Is the ratio between two corresponding sides of two triangles enough to determine their similarity?

NO

Show question

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