Types of Triangles

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

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

Examples of equilateral triangles

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

Examples of scalene triangles:

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:

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

Examples of right triangles:

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

Examples of acute triangles:

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!

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:

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

Given a right 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: $15=x$

Given the right triangle $\triangle ABC$

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

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.

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

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: $∢A>∢B$

Given the right triangle $\triangle ABC$.

$∢A=20°$

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, $∢C=70°$

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.

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 $180°$.

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 $240^o$

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

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.

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

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

$AB=AC$

$\sphericalangle B=\sphericalangle C$

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:

• Obtuse triangle
• Scalene triangle
• Equilateral triangle
• Isosceles triangle
• The edges of a triangle
• Area of a right triangle
• Height of a triangle
• How to calculate the area of a triangle
• How to calculate the perimeter of a triangle?

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