# What is a Scalene Triangle – Definition, Properties & Examples

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A triangle is a 2D shape having three sides, three vertices, and three angles. Based on the length of the sides of a triangle, it is classified into three types – equilateral, isosceles, or scalene.

Let’s understand what is a scalene triangle and what are the properties of a scalene triangle with examples.

## What is a Scalene Triangle?

A triangle in which the length of all three sides is different is called a scalene triangle. Also, the measure of the three angles of a scalene triangle is different. However, the different measurements do not affect the sum of all the interior angles of the scalene triangle. The sum of the three interior angles of a scalene triangle is always $180^{\circ}$, which satisfies the angle sum property of a triangle.

In the above figure, $\triangle \text{ABC}$ is a scalene triangle, where the measure of sides $\text{AB}$, $\text{BC}$, and $\text{CA}$ are different.

## Types of Scalene Triangles

A scalene triangle can be classified into three categories.

### Acute-Angled Scalene Triangle

An acute-angled scalene triangle is a triangle, where all three angles of a triangle are acute angles and of different measures. In the case of a scalene triangle, the circumcenter lies inside the triangle.

For example, $\triangle \text{XYZ}$  in the figure below is an acute-angled scalene triangle, where $\angle \text{X} = 60^{\circ}$, $\angle \text{Y} = 50^{\circ}$, and $\angle \text{Z} = 70^{\circ}$.

### Obtuse-Angled Scalene Triangle

An obtuse-angled scalene triangle is a triangle, one of the angles is an obtuse angle. The measure of all three angles is different in an obtuse-angled scalene triangle.  In the case of an obtuse-angled scalene triangle, the circumcenter lies outside the triangle.

For example, $\triangle \text{PQR}$  in the figure below is an obtuse-angled scalene triangle, where $\angle \text{P} = 30^{\circ}$, $\angle \text{Q} = 40^{\circ}$, and $\angle \text{R} = 110^{\circ}$.

### Right-Angled Scalene Triangle

A right-angled scalene triangle is a triangle, one of the angles is a right angle. The measure of all three angles is different in a right-angled scalene triangle.  In the case of a right-angled scalene triangle, the circumcenter lies at the midpoint of the hypotenuse.

For example, $\triangle \text{ABC}$  in the figure below is a right-angled scalene triangle, where $\angle \text{A} = 30^{\circ}$, $\angle \text{B} = 60^{\circ}$, and $\angle \text{C} = 90^{\circ}$.

## Properties of a Scalene Triangle

The following are the properties of a scalene triangle that help in distinguishing it from other types of triangles.

• A scalene triangle has three sides, each of a different length.
• In a scalene triangle, the measure of three angles is different.
• There is no line of symmetry in a scalene triangle.
• The interior angles of the triangle can be acute, obtuse, or right angles. Thus, a scalene triangle can be an obtuse triangle, an acute triangle, or a right-angled triangle.
• In a scalene obtuse triangle, the circumcenter can lie inside, outside, or on the edge of a triangle.
• In an acute-angled scalene triangle, the circumcenter lies inside a triangle.
• In a right-angled scalene triangle, the circumcenter lies on the edge (side) of a triangle. It lies on the hypotenuse.
• In an obtuse-angled scalene triangle, the circumcenter lies outside a triangle.

## Practice Problems

1. What is a scalene triangle?
2. Can the measure of angles be equal in a scalene triangle?
3. How many lines of symmetry does a scalene triangle have?
4. What is the position of the circumcenter in the case of
• acute-angled scalene triangle?
• right-angled scalene triangle?
• obtuse-angled scalene triangle?

## FAQs

### What is a scalene triangle in Geometry? A triangle in which the length of all three sides is different is called a scalene triangle. Also, the measure of the three angles of a scalene triangle is different.

### What is the difference between an isosceles triangle and a scalene triangle?

In a scalene triangle, none of the sides are of the same length and none of the angles are of equal measure. In an isosceles triangle, two out of the three sides are of the same length and the angles opposite to equal sides are of equal measure.

### Can scalene triangles be obtuse-angled? Yes, scalene triangles can be obtuse triangles. In an obtuse scalene triangle, one of the interior angles is more than $90^{\circ}$, and the other 2 angles are less than  $90^{\circ}$.

## Conclusion

A triangle in which the length of all three sides is different is called a scalene triangle. Also, the measure of the three angles of a scalene triangle is different. Since the length of all three sides of a scalene triangle is different, there is no line of symmetry in a scalene triangle.