Isosceles triangle

Definition of isosceles triangle

The isosceles triangle is a type of triangle that has two sides (legs) of equal length.

A consequence of having two sides of equal length implies that also the two angles opposite these sides measure the same.

Detailed explanation

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

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In an isosceles triangle the two equal sides are called legs and the third is called the base and in most exercises acts as the base.

The angle that lies between the two equal sides is called the angle at the vertex (or apex angle).

The two angles adjacent to the base are called base angles and measure the same.

Furthermore, the base angles that measure the same cannot be obtuse (more than $90°$ ) or right angles (equal to $50°$ ), because their measures would add up to at least 180°, therefore, they have to be acute (less than $590°$ ).

The above causes the isosceles triangle to be further classified as obtuse, right or acute, depending on how the vertex angle is.

Next, we will see some examples of isosceles triangles:

Isosceles triangle

Examples of isosceles triangles

We will demonstrate the characteristics of isosceles triangles by means of an exercise.

Given the isosceles triangle $\triangle KLM$ as shown in the figure.

Use the data shown in the illustration to calculate the angles L and M.

We will start with the triangle $\triangle KMS$ . We already know two angles, so we can calculate the third angle $M$ (The sum of the interior angles of a triangle is $180°$ degrees). Thus, the angle $M$ measures $50°$ degrees (since $50° = 180° - 100° - 30°$ ).

Since we know that the triangle $\triangle KLM$ is isosceles we understand that its base angles $L$ and $M$ are equal.

Therefore $L=M=50°$

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Test your knowledge

Question 1

Can a right triangle be equilateral?

Question 2

Choose the appropriate triangle according to the following:

Angle B equals 90 degrees.

Question 3

Does every right triangle have an angle? The other two angles are?

Questions on the subject

What is an isosceles triangle?

It is a triangle that has two sides of equal length.

What is an acute isosceles triangle?

It is an isosceles triangle whose angle at the vertex measures less than $90°$ .

What is a right isosceles triangle?

It is an isosceles triangle whose angle at the vertex measures exactly $90°$ .

If you are interested in learning more about other triangle topics, you can enter one of the following articles:

On the Tutorela blog you will find a variety of articles about mathematics.

Ejemplos y ejercicios con soluciones de triángulo isósceles

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

Choose the appropriate triangle according to the following:

Angle B equals 90 degrees.

Video Solution

Step-by-Step Solution

Let's note in which of the triangles angle B forms a right angle, meaning an angle of 90 degrees.

In answers C+D, we can see that angle B is smaller than 90 degrees.

In answer A, it is equal to 90 degrees.

Answer

Exercise #3

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

Video Solution

Step-by-Step Solution

As is known, a scalene triangle is a triangle in which each side has a different length.

According to the given information, this is indeed a triangle where each side has a different length.

Answer

Yes

Exercise #4

In a right triangle, the sum of the two non-right angles is...?

Video Solution

Step-by-Step Solution

In a right-angled triangle, there is one angle that equals 90 degrees, and the other two angles sum up to 180 degrees (sum of angles in a triangle)

Therefore, the sum of the two non-right angles is 90 degrees

$90+90=180$

Answer

90 degrees

Exercise #5

Is the triangle in the drawing a right triangle?

Step-by-Step Solution

Due to the presence of the 90 degree angle symbol we can determine that this is indeed a right-angled triangle.

Answer

Yes

Do you know what the answer is?

Question 1

Does the diagram show an obtuse triangle?

Question 2

Does the diagram show an obtuse triangle?

Question 3

Does the diagram show an obtuse triangle?

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