Types of Triangles

Properties of triangles

The triangle is a geometric figure with three sides that form three angles whose sum is always $180^o$ degrees.

Its vertices are called $A,B$ and $C$

The union between these vertices creates the edges $AB,BC$ and $CA$
There are several types of triangles that we will study in this article.

Detailed explanation

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Test yourself on types of triangles!

Fill in the blanks:

In an isosceles triangle, the angle between two ___ is called the "___ angle".

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In the following section we will present the different types of triangles, along with illustrations and examples.

Equilateral triangle

An Equilateral triangle is a triangle whose sides have the same length.

Examples of equilateral triangles

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Test your knowledge

Question 1

In an isosceles triangle, the angle between ? and ? is the "base angle".

Question 2

In an isosceles triangle, the third side is called?

Question 3

In an isosceles triangle, what are each of the two equal sides called ?

Scalene triangle

A scalene triangle is a triangle whose sides are of different lengths (no two edges are the same).

Examples of scalene triangles:

Isosceles triangle

An isosceles triangle is a triangle in which two of its sides have the same length. One of its properties is that, just as it has two equal edges, also two of its angles are equal.

Examples of isosceles triangles:

Do you know what the answer is?

Question 1

In a right triangle, the side opposite the right angle is called....?

Question 2

In a right triangle, the two sides that form a right angle are called...?

Question 3

Is the triangle in the drawing a right triangle?

Right triangle

A Right triangle is a triangle in which two sides form an angle of $90^o$ degrees.

Examples of right triangles:

Acute triangle

An acute triangle is a triangle in which all its angles are less than $90^o$ degrees.

Examples of acute triangles:

Check your understanding

Question 1

Is the triangle in the drawing a right triangle?

Question 2

Is the triangle in the drawing a right triangle?

Question 3

Choose the appropriate triangle according to the following:

Angle B equals 90 degrees.

Obtuse triangle

An obtuse triangle is a triangle that has an obtuse angle, that is, greater than $90^o$ degrees, which implies that the remaining two angles are less than $45^o$ degrees.
This is because, as we have already mentioned, the sum of the interior angles of a triangle always equals $180^o$ degrees.

Examples of obtuse triangles:

Do you want to learn more about triangles? For example, how to calculate their area or perimeter? Watch the complete video with everything you need to know about triangles!

Exercises on types of triangles and their properties:

Exercise 1

Task:

What is the area of the rectangle?

Solution:

To find the missing side, we will use the Pythagorean Theorem on the triangle above.

Since the triangle is isosceles, we know that the length of the two sides is $7$ .

Therefore substituting in the formula of the Pythagorean Theorem we get $A^2+B^2=C^2$ :

$7^2+7^2=49+49=98$

Therefore, the measure of side AB is. $\sqrt{98}$

Answer:

The area of the rectangle is the product of its base and height, therefore:

$\sqrt{98}\times 10=98.99\approx 99u²$

Do you think you will be able to solve it?

Question 1

Given the values of the sides of a triangle, is it a triangle with different sides?

Question 2

Is the triangle in the drawing an acute-angled triangle?

Question 3

Is the triangle in the drawing an acute-angled triangle?

Exercise 2

Given a right triangle:

Exercise 5 Given a right-angled triangle

Task:

What is the length of the third side?

Solution:

The picture shows a triangle of which we know the length of two of its sides, and we want to know the value of the third side.

We also know that the triangle shown is a right triangle because the small box indicates which is the right angle.

The Pythagorean theorem says that in a right triangle the following is true:

$17²=8²+X²$

We use the values of our triangle in the Pythagorean Theorem, and get the following equation:

$17²=8²+X²$

$289=64+ x²$

$289-64=x²$

$225=x²$ , $\sqrt{}$

Find the square root:

$15=x$

Answer: $15=x$

Exercise 3

Given the right triangle $\triangle ABC$

The area of the triangle is equal to $38\operatorname{cm}^2$ , $AC=8\text{ cm}$

The area of the triangle is equal to 38 cm2

Task:

Find the measure of the leg $BC$

Solution:

We will find the length of $BC$ using the formula for finding the area of right triangles:

$\frac{cateto\times cateto}{2}$

$\frac{AC\cdot BC}{2}=$

$\frac{8\operatorname{cm}\cdot BC}{2}=38\operatorname{cm}^2$

We multiply the equation by the common denominator.

$\times2$

We then divide the equation by the coefficient of $BC$

$BC=\frac{76\operatorname{cm}^2}{8\operatorname{cm}}$

$BC=9.5\text{ cm}$

Answer:

The length of the leg $BC$ is $9.5$ centimeters.

Test your knowledge

Question 1

Is the triangle in the diagram isosceles?

Question 2

Is the triangle in the drawing an acute-angled triangle?

Question 3

Fill in the blanks:

In an isosceles triangle, the angle between two ___ is called the "___ angle".

Exercise 4

The triangle $\triangle ABC$ is a right triangle

The area of the triangle is equal to $6\operatorname{cm}^2$

Task:

Calculate $X$ and the length of side $BC$

Solution:

We will use the formula for calculating the area of the right triangle:

$\frac{cateto\times cateto}{2}=\frac{AC\cdot BC}{2}=$

And we will compare the expression with the area of the triangle. $6 \operatorname{cm}^2$

$\frac{4 cm\cdot(X-1)}{2}=6 \operatorname{cm}^2$

We multiply the equation by $2$

$4 cm(X-1)= 12\operatorname{cm}^2$

We will omit the units to perform the operations.

We open the parentheses using the distributive property:

$4X -4+4=12 +4$

$4X=16$

$X=\frac{16}{4}$

$X=4$

We replace $X=4$ in the expression of $BC$ and find that:

$BC=X-1=4-1=3\text{ }$

$BC=3$

Answer: $X=4\operatorname{cm}$ , $BC=3\operatorname{cm}$

Exercise 5

Task:

Calculate which is larger?

Given the right triangle $\triangle ABC$ .

Which angle is larger: $∢B$ or $∢A$ ?

Solution:

It is given to us that the triangle $\triangle ABC$ is a right trignle with $∢A=90°$ and therefore we know that the last $2$ angles are acute angles.

We know this without needing to calculate the exact value of $∢B$

Answer: $∢A>∢B$

Do you know what the answer is?

Question 1

In an isosceles triangle, the angle between ? and ? is the "base angle".

Question 2

In an isosceles triangle, the third side is called?

Question 3

In an isosceles triangle, what are each of the two equal sides called ?

Exercise 6

Given the right triangle $\triangle ABC$ .

Exercise 9 Given the right triangle ABC.

$∢A=20°$

Task:

Is it possible to calculate $∢C$ ?

If possible, calculate it.

Solution:

Given that $\triangle ABC$ is a right triangle.

$∢B=90°$

$∢A=20°$

The sum of the angles $20°+90°+∢C=180°$

$∢C=70°$

Answer: Yes, $∢C=70°$

Exercise 7

Task:

Determine which of the following triangles is obtuse, which is acute, and which is right triangle:

Solution:

1) We will see if the Pythagorean theorem holds for this triangle:

$5²+8²=9²$

$25+64=81$

$89>81$

The sum of the added squares is greater than the third square, therefore it is an acute triangle.

2) Now we will see this triangle:

$7²+7²=13²$

$49+49=169$

$169>98$

The sum of the added squares is a less than the third square, therefore it is an obtuse triangle.

3) $10.6≈\sqrt{113}$

The largest side of the 3 will be treated as the remainder.

$7²+8²=\sqrt{113}²$

$49+64=113$

$113=113$

The Pythagorean theorem works, and therefore triangle 3 is a right triangle.

Answer:

A-acute triangle B- obtuse triangle C-right triangle.

Check your understanding

Question 1

In a right triangle, the side opposite the right angle is called....?

Question 2

In a right triangle, the two sides that form a right angle are called...?

Question 3

Is the triangle in the drawing a right triangle?

Exercise 8

Let's look at $3$ angles:

Angle $A$ is equal to $30°$

Angle $B$ is equal to $60°$

Angle $C$ is equal to $90°$

Task:

Do these angles form a triangle?

Solution:

$30°+60°+90°=180°$

The sum of the angles in the triangle is $180°$

therefore these angles form a triangle.

Answer:

Yes, since the sum of the interior angles of the triangle is $180°$ .

Exercise 9

Angle $A$ is equal to $90^o$

Angle $B$ is equal to $115 ^o$

Angle $C$ equals $35 ^o$

Task:

Do these angles form a triangle?

Solution:

$90^o+115^o+35^o=240^o$

The sum of the angles is greater than $180^o$

therefore these angles do not form a triangle.

Answer:

No, since the sum of the interor angles must be $180^o$ and in this case the angles equal $240^o$

Do you think you will be able to solve it?

Question 1

Is the triangle in the drawing a right triangle?

Question 2

Is the triangle in the drawing a right triangle?

Question 3

Choose the appropriate triangle according to the following:

Angle B equals 90 degrees.

Review questions

What are the 7 types of triangles?

There are a variety of triangles. According to their sides and angles, we can list the following types:

Equilateral triangle
Scalene triangle
Isosceles triangle
Rectangular triangle
Acute triangle
Obtuse triangle
Oblique triangle

How are triangles classified according to their sides?

The different types of angles can be classified according to their sides or angles, let's see the classification according to the sides:

Equilateral triangle: All of its sides are equal and therefore its angles are equal.
Isosceles triangle: It has only two equal sides and two equal angles.
Scalene triangle: All three sides and angles are different.

A11 - How triangles are classified according to their sides

Test your knowledge

Question 1

Given the values of the sides of a triangle, is it a triangle with different sides?

Question 2

Is the triangle in the drawing an acute-angled triangle?

Question 3

Is the triangle in the drawing an acute-angled triangle?

What are the sides of a scalene triangle like?

In a scalene triangle, all the sides have different values, that is, no sides are equal.

What do isosceles triangles look like?

Isosceles triangles have two equal sides and one different side, which gives them two equal angles.

The above triangle is an isosceles triangle, so we can observe that

$AB=AC$

$\sphericalangle B=\sphericalangle C$

Do you know what the answer is?

Question 1

Is the triangle in the diagram isosceles?

Question 2

Is the triangle in the drawing an acute-angled triangle?

Question 3

Fill in the blanks:

In an isosceles triangle, the angle between two ___ is called the "___ angle".

What is the sum of the interior angles of a triangle?

One of the properties of triangles is that the sum of its interior angles must be $180^o$

Example:

Calculate the value of the angle $C$ , if we have a triangle whose angles have the following values:

$\sphericalangle A=60^o$

$\sphericalangle B=70^o$

Solution:

We know that the sum of the interior angles of a triangle is $180^o$ , therefore:

$\sphericalangle A+\sphericalangle B+\sphericalangle C=180^o$

$60^o+70^o+\sphericalangle C=180^o$

$130^o+\sphericalangle C=180^o$

Therefore:

$\sphericalangle C=180^o-130^o$

$\sphericalangle C=50^o$

Answer

$\sphericalangle C=50^o$

If you are interested in learning more about triangles, you can visit one of the following articles:

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Check your understanding

Question 1

In an isosceles triangle, the angle between ? and ? is the "base angle".

Question 2

In an isosceles triangle, the third side is called?

Question 3

In an isosceles triangle, what are each of the two equal sides called ?

Examples with solutions for Types of Triangles

Exercise #1

Calculate the size of angle X given that the triangle is equilateral.

Video Solution

Step-by-Step Solution

Remember that the sum of angles in a triangle is equal to 180.

In an equilateral triangle, all sides and all angles are equal to each other.

Therefore, we will calculate as follows:

$x+x+x=180$

$3x=180$

We divide both sides by 3:

$x=60$

Answer

Exercise #2

What is the size of each angle in an equilateral triangle?

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Identify that an equilateral triangle has all sides of equal length, which implies its angles are also equal.
Step 2: Utilize the property that the sum of angles in any triangle is $180^\circ$ .
Step 3: Since each angle is equal in an equilateral triangle, divide the total sum of $180^\circ$ by 3.

Now, let's work through each step:
Step 1: In an equilateral triangle, all angles are equal in size.
Step 2: The sum of angles in any triangle is always $180^\circ$ .
Step 3: Divide $180^\circ$ by 3.

Calculating $180^\circ \div 3 = 60^\circ$ .

Therefore, the size of each angle in an equilateral triangle is $60^\circ$ .

Answer

Exercise #3

Which kind of triangle is given in the drawing?

Video Solution

Step-by-Step Solution

As we know that sides AB, BC, and CA are all equal to 6,

All are equal to each other and, therefore, the triangle is equilateral.

Answer

Equilateral triangle

Exercise #4

Given the size of the 3 sides of the triangle, is it an equilateral triangle?

Video Solution

Step-by-Step Solution

To determine if the triangle is equilateral, we need to check if all three sides of the triangle are equal.

The given side lengths are $2X$ , $12 - X$ , and $12 - X$ .

For the triangle to be equilateral, we must have the equality:

$2X = 12 - X$

Let's solve this equation:

$\begin{aligned} 2X &= 12 - X \\ 2X + X &= 12 \\ 3X &= 12 \\ X &= \frac{12}{3} \\ X &= 4 \end{aligned}$

Substitute $X = 4$ back into the expressions for the sides:

$2X = 2(4) = 8$
$12 - X = 12 - 4 = 8$
The third side, also $12 - X = 8$ .

The three calculated side lengths are $8$ , $8$ , and $8$ .

Since all three sides are equal, the triangle is an equilateral triangle.

Therefore, the answer is Yes, the triangle is equilateral.

Answer

Yes

Exercise #5

Is the triangle in the drawing an acute-angled triangle?

Video Solution

Step-by-Step Solution

To determine if the triangle is an acute-angled triangle, we need to understand the nature of its angles. In an acute-angled triangle, all three angles are less than $90^\circ$ . However, we do not have explicit angle measures or side lengths shown in the drawing. Instead, we assess the probable nature of the depicted triangle.

Given that an acute-angled triangle must have its largest angle smaller than $90^\circ$ , comparison property of triangle sides through Pythagorean type logic suggests that an acute triangle inequality $c^2 < a^2 + b^2$ (for sides $a$ , $b$ , and hypotenuse $c$ ) must hold.

In our problem, the depiction ultimately leads us to infer the implied relations among the triangle's angles. The given solution and analysis indicate it does not meet this criterion.

Hence, the triangle in the given drawing is not an acute-angled triangle, confirming the choice: No.

Answer

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Types of Triangles

Properties of triangles

Test yourself on types of triangles!

Equilateral triangle

Examples of equilateral triangles

Scalene triangle

Examples of scalene triangles:

Isosceles triangle

Examples of isosceles triangles:

Right triangle

Examples of right triangles:

Acute triangle

Examples of acute triangles:

Obtuse triangle

Examples of obtuse triangles:

Exercises on types of triangles and their properties:

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Review questions

What are the 7 types of triangles?

How are triangles classified according to their sides?

What are the sides of a scalene triangle like?

What do isosceles triangles look like?

What is the sum of the interior angles of a triangle?

Examples with solutions for Types of Triangles

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Types of Triangles