### Edexcel GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

# Triangles

A triangle is a polygon with three sides and three angles. It is formed by connecting three points in a flat space that are not in a straight line. These points, called vertices, are connected by straight lines, which form the sides of the triangle.

## Types of Triangles

Triangles can be categorised into three types based on their sides:

• Scalene Triangle – A triangle with all sides of different lengths. For example, a triangle with sides of 3 cm, 4 cm, and 5 cm is a scalene triangle. Also, none of the angles are equal in a scalene triangle.
• Isosceles Triangle – A triangle with two sides of equal length. For example, a triangle with sides of 5 cm, 5 cm, and 3 cm is an isosceles triangle. The angles opposite the equal sides are also equal.
• Equilateral Triangle – A triangle with all sides equal in length. For example, a triangle with sides of 4 cm each is an equilateral triangle.

### Classification by Angles

Triangles can also be classified based on their angles:

• Acute Triangle – A triangle with all three angles less than 90°. For example, a triangle with angles measuring 35°, 60° and 85° is an acute triangle.
• Right-angle Triangle – A triangle with one angle measuring exactly 90°. For example, a triangle with angles measuring 90°, 30° and 60° is a right triangle.
• Obtuse Triangle – A triangle with one angle greater than 90°. For example, a triangle with angles measuring 110°, 40° and 30° is an obtuse triangle.

We can also identify an equilateral triangle if all three angles inside the triangle are equal to each other. All angles in an equilateral triangle measure 60°.

## Labelling Sides and Angles in Triangles

To solve problems related to triangles, it’s important to know how to label their sides and angles.

First, label the three vertices with capital letters, such as A, B, and C.

You can represent the sides by combining the vertex letters. For example, side AB is formed by connecting vertices A and B and side BC is formed by connecting vertices B and C.

Also, you can refer to the angles by the sequence of the vertices they span. In the diagram above, the angle between lines AB and AC is labelled BAC.

## Examples

Example: Identifying Types of Triangles

Classify the following triangles based on their sides and angles:

a) A triangle with sides of 3 cm, 4 cm and 5 cm.

b) A triangle with angles measuring 60°, 60° and 60°.

c) A triangle with angles measuring 90°, 45° and 45°.

a) Scalene triangle

As all sides have different lengths, it is also a scalene triangle. Therefore, the triangle is a scalene right-angled triangle. It is also a right-angled triangle since the side lengths follow the Pythagorean theorem (3² + 4² = 5²).

b) Equilateral and acute triangle

Since all the angles are equal, this triangle is an equilateral triangle. Equilateral triangles also have all angles equal to 60°, making them acute-angled.

c) Isosceles and right-angled triangle

Since one angle measures 90°, this triangle is a right-angled triangle. The other two angles are equal (45° each), making the triangle isosceles.

Quizzes