## Introduction

In the variety of geometrical shapes we encounter daily, have you ever wondered how to relate two shapes about their size and orientation? Turns out, mathematicians have made an answer to this—the concept of similarity.

In this article, we introduce this concept by comparing triangles. We first look back at the definition and properties of a triangle with some relevant equations, then we show different theorems associated with similar triangles. Along the way, we will also try some examples as a supplement for learning.

### What is a Triangle?

In geometry, a triangle is a shape that consists of a set of three straight lines or **sides**, three **interior angles **that are formed by a pair of sides, and three **vertices** or intersection points of the lines forming the triangle.

A triangle can also have an **exterior angle**. This occurs when we extend one side of a triangle and take the angle the extended line forms at its vertex with another side.

### What are some Properties of Triangles?

First, we note that the sum of the length of two sides of a triangle is greater than the length of its third side. Conversely, we can say that the difference between the length of two sides is always less than the length of the third side.

Second, we observe that for any triangle the sum of its interior angles adds up to a measure of 180°. This is called the **Angle Sum Property**.

Third, we note that the sum of the measures of a triangle’s interior angle and its adjacent exterior angle is supplementary. In other words, these angles add up to a measure of 180°.

### What is the Law of Sines?

The **Law of Sines**, also known by other names such as Sine Law, Sine Rule, or Sine Formula, is an equation relating the angles of a triangle with the corresponding side opposite its vertex.

It states that the ratio between the length of a triangle’s side and the sine of the angle at an opposite vertex is equal for all sides and angles of any triangle:

$\frac{a}{sinsin A} = \frac{b}{sinsin B} = \frac{c}{sinsin C}$

We can use the Law of Sines to find the unknown sides or angles of a triangle, through a set of given information:

- Two angles of a triangle and one included side
- Two angles of a triangle and one non-included side
- Two sides of a triangle and one angle opposite either side

### What is the Law of Cosines?

The **Law of Cosines**, also called the Cosine Law, Cosine Rule, or Cosine Formula, is another equation that relates the angles of a triangle with the lengths of its sides.

It states that the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, minus twice their product and the cosine of their included angle:

c^{2} = a^{2 }+ b^{2} – 2ab coscos C

Alternatively, we can write similar equations for the remaining two sides:

a^{2} = b^{2} + c^{2} – 2ab coscos A

b^{2} = a^{2} + c^{2} – 2ac coscos B

We can use the Law of Cosines to find the following unknown:

- Unknown side of a triangle given two sides and one included angle
- Unknown angles of a triangle given all three sides

## Similar Triangles

Now, we move from the underlying concepts to the main topic. **Similarity**, by definition, is the **likeliness or resemblance** of two geometrical objects. When we say two objects are similar, we observe that they look alike. Whether it is by their size or orientation, we recognize how we get from one shape to another in similar shapes.

For the case of triangles, we say that two triangles are similar if their sizes are proportional and/or one triangle is either a rotated or flipped version of the other:

To be specific, two triangles ABC and XYZ are said to be **similar** if the following conditions hold:

- The lengths of their sides are proportional to each other

$\frac{AB}{XY} = \frac{AC}{XZ} = \frac{BC}{YZ}$

- The measures of their corresponding angles are equal

∠A≅∠X

∠B≅∠Y

∠C≅∠Z

In the above example, we have triangles ABC and XYZ which are scaled, rotated versions of each other. This means that they are similar triangles, or in mathematical notation: ABC∼XYZ where “” is the symbol denoting similarity.

With these conditions, we have the following theorems that involve similarity between triangles: the Angle-Angle Similarity Theorem, the Side-Angle-Side Similarity Theorem, and the Side-Side-Side Similarity Theorem.

### AA Similarity Theorem

The **Angle-Angle (AA) Similarity Theorem** determines similar triangles based on a pair of two angles in triangles. It states that if the measure of two angles of a triangle is equal to the measure of two angles in another triangle, then the two triangles are similar.

This also refers to the Angle-Angle-Angle (AAA) Similarity Theorem, since by the Angle Sum Property we can tell that the third angle and its corresponding angle are equal.

### SAS Similarity Theorem

The **Side-Angle-Side (SAS) Similarity Theorem** determines the similarity between two triangles by comparing a set of two angles and the angle formed between these sides.

It states that if the measure of the lengths of any two sides in a triangle is proportional to the length of two sides in another triangle, and the measures of the included angles in both triangles are equal, then they are similar triangles.

### SSS Similarity Theorem

The **Side-Side-Side (SSS) Similarity Theorem **checks whether two triangles are similar based on the sides of both triangles being compared.

It states that if the measures of the lengths of all sides of a triangle are proportional to the lengths of the sides of the other triangle, then they are similar.

## Problem-Solving Examples

We can now proceed to solve sample problems to apply what we have learned so far. Each problem tackles different formulas discussed and gives us a challenge on how to solve through the information given to us.

### Using the AA Similarity Theorem

**Sample Problem 1:**

In the figure below, we have two triangles FGH and JKL. Are the two triangles similar?

**Solution:**

We first note the given measures of each angle:

∠F=40° ∠G=80° ∠H=60°

∠J=80° ∠K=60° ∠L=40°

We then relate the corresponding angles of both triangles to establish the following equalities:

∠F≅∠L

∠G≅∠J

∠H≅∠K

Now, since at least two pairs of angles have the same measure, we conclude that by the AA Similarity Theorem the given triangles are **similar**:

△FGH∼△JKL

**Sample Problem 2:**

Two triangles CAT and DOG whose angles are given are shown below:

We are asked the following questions:

- What is the measure of the unknown angle ∠C?
- What is the measure of the unknown angle ∠D?
- Are CAT and DOG similar triangles?

**Solution:**

- We first consider triangle CAT. Two of its angles are given with measures:

∠A=35°

∠T=75°

Then, by the Angle Sum Property we can relate the sum of the interior angles of CAT as:

∠C+∠A+∠T=180°

Substituting the given angle measures, we have:

∠C+35°+75°=180°

Next, we rewrite the equation in terms of the unknown angle by transposing terms to the right-hand side of the equation:

∠C=180°-35°-75°

Finally, we subtract the measures of the angles to get the measure of the unknown angle:

∠C=70°

Therefore, the measure of ∠C is 70°.

- Following the steps performed in Part A, we consider the other triangle DOG and note the given measures of its angles:

∠O=70°

∠G=75°

Then, by the Angle Sum Property we can relate the measures of the interior angles of DOG as follows:

∠D+∠O+∠G=180°

Substituting the given angle measures, we get:

∠D+∠70°+∠75°=180°

We then rewrite the equation in terms of the unknown angle:

∠D=180°-70°-75°

Lastly, we subtract the given measures to obtain the measure of the unknown angle:

∠D=35°

Therefore, the measure of ∠D is 35°.

- From the angle measures obtained from Part A and Part B, we relate the corresponding angles of both triangles to establish the following equalities:

∠C≅∠O

∠A≅∠D

∠T≅∠G

Now, since at least two pairs of angles have the same measure, we conclude that by the AA Similarity Theorem the given triangles are **similar**:

△CAT∼△DOG

**Sample Problem 3:**

Suppose we have two triangles △ELF and △ORC whose sides and angles are given as shown below:

Show that the two triangles are similar via the AA Similarity Theorem.

**Solution:**

To use the AA Similarity Theorem, we must show that at least two pairs of angles in △ELF and △ORC are equal in measure. However, only one angle is given for triangle ELF. Hence, we first solve for the measure of either ∠L or ∠F.

In this case, we choose to solve for ∠L using the Law of Sines. We recall that for triangle ELF, the following equality holds following the Sine Law:

$\frac{e}{sinsin E} = \frac{l}{sinsin L} = \frac{f}{sinsin F}$

Focusing on the given angle and side lengths, we have:

$\frac{8}{sinsin 75°} = \frac{7}{sinsin L}$

Re-writing the equation in terms of the unknown angle measure L, we get:

sinsin L = $\frac{7}{8}$ x sinsin 75°

Solving for the value of sin 75° , we simplify the right-hand side of the equation into:

sinsin L= $\frac{7}{8}$ × 0.97

Then, we can multiply the numbers together to obtain:

sinsin L=0.85

Finally, we take the arcsine of both sides of the equation to solve for the measure of ∠L:

L=arcsin arcsin (0.85)

∴L=58°

Now, we observe that for the given triangles we have the following congruent angles:

∠E≅∠C

∠L≅∠R

Therefore, by the AA Similarity Theorem we conclude that both triangles are **similar**:

△ELF∼△ORC

### Using the SAS Similarity Theorem

**Sample Problem 4:**

We consider the same pair of triangles from Sample Problem 3:

This time, show that △ELF∼△ORC using the SAS Similarity Theorem.

**Solution:**

From the given figure, we know the angle measures of both triangles to be:

∠E=75°

∠O=47°

∠R=58°

From the given angles, we note that the measure of angles ∠E and ∠C are equal:

∠E≅∠C

Next, we observe the given lengths of the sides of both triangles:

EF=7 LF=8

OC=21 OR=24

If we consider the side opposite to the congruent angles, and one side adjacent to it, we compare the ratio of their measures between triangles △ELF and △ORC, we get:

$\frac{EF}{OC} = \frac{7}{21} = \frac{1}{3}$

$\frac{LF}{OR} = \frac{8}{24} = \frac{1}{3}$

Hence, we can say that the two sides of both triangles are proportional to each other:

$\frac{EF}{OC} = \frac{LF}{OR}$

Since ∠E≅∠C, and $\frac{EF}{OC} = \frac{LF}{OR}$, by the SAS Similarity Theorem we therefore conclude that the given triangles are **similar**:

△ELF ∼ △ORC

**Sample Problem 5:**

In the figure shown below, we have two isosceles triangles whose sides and angles are known:

Are the two triangles similar?

**Solution:**

We first note that the given triangles are isosceles. This means that the lengths of two sides are equal, and the measures of two internal angles are also equal. In this case, we have the following information given to us:

∠E=∠R=65°

RD=ED=13

TI=TN=26

We wish to find the measure of the unknown angle ∠D. By the Angle Sum Property, we can relate the internal angles of RED such that:

∠R+∠E+∠D=180°

Substituting the given angle measures, we have:

65°+65°+∠D=180°

Then, we transpose terms to isolate the unknown angle ∠D:

∠D=180°-65°-65°

Hence, we can determine the measure of angle ∠D by subtracting the numbers on the right-hand side of the equation:

∠D=50°

We note that this angle has a measure equal to that of ∠T. As such, we can say that both angles are congruent:

∠D≅∠T

Next, if we consider the equal sides of both triangles and take their ratio, we get the following equalities:

$\frac{RD}{TI}=\frac{13}{26}=\frac{1}{2}$

$\frac{ED}{TN}=\frac{13}{26}=\frac{1}{2}$

Thus, we can say that the two sides of both triangles are proportional to each other:

$\frac{RD}{TI}=\frac{ED}{TN}$

Finally, since ∠D≅∠T, and $\frac{RD}{TI}=\frac{ED}{TN}$, we therefore conclude by the SAS Similarity Theorem that both triangles are **similar**:

△RED ∼ △TIN

### Using the SSS Similarity Theorem

**Sample Problem 6:**

Suppose we have two equilateral triangles △ABC and △XYZ:

- If their side lengths are given by AB=1 and XY=10, are the two triangles similar?
- If the side lengths of △ABC are changed to AB=20 and △XYZ remains unchanged, are they still similar triangles?
- In general, can we say that all equilateral triangles are similar? Why or why not?

**Solution:**

- We recall that for equilateral triangles, all sides have equal length. This implies that the side lengths for both triangles are given as:

AB=AC=BC=1

XY=XZ=YZ=10

Now, if we take the ratio between corresponding sides of both triangles, we observe that they have the same value:

$\frac{AB}{XY}=\frac{1}{10}$

$\frac{AC}{XZ}=\frac{1}{10}$

$\frac{BC}{YZ}=\frac{1}{10}$

Hence, we can say that all sides of both triangles are proportional to each other:

$\frac{AB}{XY}=\frac{AC}{XZ}=\frac{BC}{YZ}$

Therefore, by the SSS Similarity Theorem we conclude that the given triangles are **similar**:

△ABC∼△XYZ

- We repeat the same steps done in Part A. Since the given lengths are:

AB=AC=BC=20

XY=XZ=YZ=10

We know that the ratio between their corresponding sides is equal:

$\frac{AB}{XY}=\frac{AC}{XZ}=\frac{BC}{YZ}=\frac{20}{10}$

Thus, we conclude by the SSS Similarity Theorem that the two triangles are also **similar**:

△ABC∼△XYZ

- We have observed that changing the side lengths of either triangle still preserves similarity between the two triangles. Since the side lengths of one triangle change in proportion to its other sides, we expect the ratio between the side lengths of both triangles to remain equal.

Through this observation, we can say that **all equilateral triangles are similar**.

**Sample Problem 7:**

Two triangles △RST and △UVW are shown whose side lengths and angle measures are given:

- What is the length of side RS of triangle △RST?
- Are the two triangles similar?

**Solution:**

- We observe that for triangle △RST, two sides and their included angle are given. We need to know all three sides of both triangles to use the SSS Similarity Theorem. Hence, we can apply the Law of Cosines to solve for the length of side RS.

We recall from the Cosine Law that the lengths of the sides of △RST can be expressed as follows:

RS^{2}=ST^{2}+RT^{2} – (2×ST×RT×coscos T )

Substituting the given side lengths and included angle, we have:

RS^{2}=16^{2}+14^{2}-(2×16×14×coscos 47°)

We then evaluate the value of the trigonometric function:

RS^{2}=16^{2}+14^{2}-(2×16×14×0.682)

Next, we multiply the numbers enclosed within parentheses:

RS^{2}=16^{2}+14^{2}-305.536

We also take the square of both numbers on the right-hand side of the equation:

RS^{2}=256+196-304.64

Thus, we can simplify the equation by adding/subtracting the numbers together:

RS^{2}=146.464

We take the square root of both sides to get the length of side RS:

RS=12.102

Rounding off to the nearest whole number, we conclude that the length of RS is 12:

RS=12

- Now that we know the side lengths of both triangles to be:

RS=12 | RT=14 | ST=16 |

VW=36 | UW=42 | UV=48 |

If we take the ratio between the corresponding sides of both triangles, we have the following values:

$\frac{RS}{VW}=\frac{12}{36}=\frac{1}{3}$

$\frac{RT}{UW}=\frac{14}{42}=\frac{1}{3}$

$\frac{ST}{UV}=\frac{16}{48}=\frac{1}{3}$

Thus, we can say that all sides of both RST and UVW are proportional to each other:

$\frac{RS}{VW}=\frac{RT}{UW}=\frac{ST}{UV}$

Therefore, by the SSS Similarity Theorem we conclude that both triangles are **similar**:

△RST∼△UVW

**Sample Problem 8:**

In the figure shown below, we have two right triangles △LMN and △OPQ:

- What is the length of the hypotenuse of △LMN?
- What is the length of side OP of △OPQ?
- Are the given triangles similar?

**Solution:**

- We begin by solving for the length of the hypotenuse of △LMN by applying the Pythagorean Theorem. We recall for a right triangle, the length of the hypotenuse is given by:

LN^{2}=LM^{2}+MN^{2}

Substituting the given side lengths, we have:

LN^{2}=8^{2}+6^{2}

We then take the square of both numbers we have substituted earlier to get:

LN^{2}=64+36

Then, we add both numbers on the right-hand side of the equation:

LN^{2}=100

Lastly, we take the square root of both sides of the equation to get the length of the hypotenuse:

LN=10

Therefore, the length of the hypotenuse LN is 10.

- We then proceed to solve for the length of the unknown side of △OPQ through the Pythagorean Theorem. Again, for a right triangle, their side lengths are related as:

OQ^{2}=OP^{2}+PQ^{2}

Re-writing the equation in terms of the unknown side OP, we get:

OP^{2}=OQ^{2}-PQ^{2}

Substituting the given side lengths, we have:

OP^{2}=30^{2}-18^{2}

We then take the square of both numbers we have substituted earlier to get:

OP^{2}=900-324

Subtracting both numbers on the right-hand side of the equation, we have:

OP^{2}=576

Finally, we take the square root of both sides of the equation to get the length of the unknown side OP:

OP=24

Therefore, the length of side OP is 24.

- From the computations made in Part A and Part B, we know the side lengths of both triangles to be:

LM=8 | MN=6 | LN=10 |

OP=24 | PQ=18 | OQ=30 |

If we take the ratio between the corresponding sides of both triangles, we have the following values:

$\frac{LM}{OP}=\frac{8}{24}=\frac{1}{3}$

$\frac{MN}{PQ}=\frac{6}{18}=\frac{1}{3}$

$\frac{LN}{OQ}=\frac{10}{30}=\frac{1}{3}$

Thus, we can say that all sides of both LMN and OPQ are proportional to each other:

$\frac{LM}{OP}=\frac{MN}{PQ}=\frac{LN}{OQ}$

Therefore, by the SSS Similarity Theorem we conclude that both triangles are **similar**:

△LMN∼△OPQ

## Summary

A **triangle **is a shape formed by a set of **three sides**, **three interior angles**, formed between the sides of the triangle, and **three vertices**, or intersections of the sides of the triangle. It can also have **exterior angles**, formed by extending one of its sides.

The **Angle Sum Property **states that the sum of all interior angles in a triangle adds up to a measure of 180°.

The **Law of Sines** states that the ratio between the length of a triangle’s side and the sine of the angle at an opposite vertex is equal for all sides and angles of any triangle:

$\frac{a}{sinsin A} = \frac{b}{sinsin B} = \frac{c}{sinsin C}$

On the other hand, the **Law of Cosines** relates the squares of the lengths of sides of a triangle and the cosine of their included angles:

c^{2}=a^{2}+b^{2} – 2ab coscos C

Two triangles are said to be **similar** if the following conditions hold:

The lengths of their sides are proportional to each other

The measures of their corresponding angles are equal

To determine whether two triangles are similar, we use the following similarity theorems:

The **Angle-Angle (AA) Similarity Theorem** states that if the measure of two angles of a triangle is equal to the measure of two angles in another triangle, then the two triangles are similar.

The **Side-Angle-Side (SAS) Similarity Theorem** states that if the measure of the lengths of any two sides in a triangle is proportional to the length of two sides in another triangle, and the measures of the included angles in both triangles are equal, then they are similar triangles.

The **Side-Side-Side (SSS) Similarity Theorem **states that if the measures of the lengths of all sides of a triangle are proportional to the lengths of the sides of the other triangle, then they are similar.

## Frequently Asked Questions (FAQs)

### What is an included side?

An included side is a side that coincides with two angles. For example, for a triangle FGH, the side $\overline{HF}$ is an included side between two angles ∠H and angle ∠F:

We can also say from this figure that $\overline{HG}$ is an included side between ∠H and ∠G, and $\overline{GF}$ is an included side between ∠G and ∠F.

### What is an included angle?

An included angle is an angle that coincides with two sides. Using the previous example, the angle ∠H is an included angle between two sides $\overline{HF}$ and $\overline{HG}$:

We can also say from this figure that ∠G is an included angle between $\overline{HG}$ and $\overline{GF}$, and ∠F is an included angle between $\overline{HF}$ and $\overline{GF}$.

### What is a non-included side?

A non-included side is a side that does not coincide with two angles. For example, for a triangle △FGH, the sides $\overline{HG}$ and $\overline{GF}$ are non-included sides between two angles ∠H and angle ∠F:

We can also say from this figure that $\overline{HF}$ and $\overline{GF}$ are non-included sides between ∠H and ∠G, and that $\overline{HG}$ and $\overline{HF}$ are non-included sides between ∠G and ∠F.

### What is a non-included angle?

A non-included angle is an angle that does not coincide with two sides. Using the previous example, the angles G and ∠F are non-included angles between two sides $\overline{HF}$ and $\overline{GF}$:

We can also say from this figure that H and ∠F are non-included angles between $\overline{HG}$ and $\overline{GF}$, and that H and ∠G are non-included angles between $\overline{HF}$ and $\overline{GF}$.

### What is the Pythagorean Theorem?

The Pythagorean Theorem relates all three sides of a right triangle in an equation. It states that the square of the length of its hypotenuse is equal to the sum of the squares of the lengths of its other sides:

c^{2}=a^{2}+b^{2}

### How do we use the Pythagorean Theorem?

If we know the lengths of the sides of a right triangle, we substitute their lengths to the formula. Then, we square their lengths and add them to get the squared length of the hypotenuse. After taking the square root of the sum, we then get the length of the hypotenuse c.

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