A triangle is a form having three sides and three corners. Each side links to two adjacent sides, resulting in three corners where the sides meet. The angles within a triangle always sum to 180 degrees. Triangles are classified into three types, namely equilateral (all sides and angles are equal), isosceles (two sides and two angles are equal), and scalene (all sides and angles differ).

In this article, we will understand the concepts related to triangles: meaning and definition of tringles, properties of triangles, types of triangles and formulas of triangles.

What is a Triangle?

Triangle is a basic geometric form that has three sides and three angles. Imagine connecting three points on a level area using straight lines to make a triangle. Each side of the triangle links two vertices, and an angle is generated when two sides intersect.

Consider a piece of pizza that has three sides and three corners, similar to a triangle. Another example is a street sign with three edges that intersect at each corner. Triangles are fundamental forms seen in many common items, and they are required in geometry to comprehend the basic concepts of angles and measures.

Triangle Definition

A triangle is a fundamental geometric form consisting of three straight sides and three angles. The sides meet to form three corners. The total of the angles within a triangle is always 180 degrees. Triangles have varied qualities; for example, they can be equilateral (all sides and angles equal), isosceles (two sides and angles equal), or scalene.

What are Properties of Triangle?

Before getting into the properties of triangles, it’s important to understand their basic structure and characteristics:

• A triangle is composed of three sides, three angles, and three vertices.
• Sum of all internal angles is always 180°.
• Sum of the lengths of any two sides exceeds the length of the third.
• Longest side corresponds to the biggest angle.
• Exterior angle is equal to sum of opposite two interior angle.

Types of Triangles

Triangles are classified according to their side lengths and angles. Here are the six primary categories explained simply:

Equilateral Triangle: An equilateral triangle has all sides of equal length and angles of 60 degrees. For example, consider a triangle with all three sides measuring five units long.

Isosceles Triangle: An isosceles triangle has two sides of equal length and opposing angles. For example, consider a triangle with two sides measuring four units apiece and a third side measuring six units.

Scalene Triangle: A scalene triangle has sides of varying lengths and angles of varying degrees. Consider a triangle with sides of 3, 4, and 5 units.

Acute Triangle: Acute triangles have all angles smaller than 90 degrees. For example, a triangle with angles of 40, 60, and 80 degrees.

Right Triangle: A right triangle has one angle that measures exactly 90 degrees, which is known as the right angle. Consider a triangle with angles of 30, 60, and 90 degrees.

Obtuse Triangle: An obtuse triangle has one angle greater than 90 degrees. For example, a triangle with angles measuring 20, 40, and 120 degrees.

Types of Triangles

Properties of Triangle

Properties of triangles are fundamental rules in geometry. They include the angle sum property, triangle inequality property, Pythagoras theorem, side-angle relationship, exterior angles, and congruence conditions, etc.

Some of the important properties of triangle are added below:

Angle Sum Property

Angle Sum Property is a fundamental property in geometry that asserts that the sum of all angles within a triangle is always 180 degrees. This technique is useful for solving for missing angles or determining triangle validity. For example, if two angles are 60 degrees each, the third angle must also be 60 degrees to meet this criterion.

Angle 1 + Angle 2 + Angle 3 = 180^∘

Triangle Inequality Property

The total of any two sides of a triangle exceeds the length of the third side. In other words, the shortest path between two places is a straight line. This is expressed as:

a + b > c

where a, b, and c are the lengths of the sides of the triangle.

Pythagoras Property

In a right triangle, the square of the hypotenuse’s length (the side opposite the right angle) equals the sum of the squares of the other two sides. This is called the Pythagorean theorem.

c² = a² + b²

Hypotenuse length is denoted by c, whereas the other two sides’ lengths are denoted by a and b.

Side Opposite the Greater Angle is the Longest Side

The side opposite the greatest angle in a triangle is the longest side. This is an observable property, not a formal theorem. When given a triangle’s angles, it helps to determine which side is the longest.

Exterior Angle Property

Each exterior angle of a triangle equals the sum of its two remote interior angles. The mathematical expression is:

Exterior Angle = Sum of Remote Interior Angles

Congruence Property

Triangles of the same size and shape are said to be congruent. This attribute is useful in assessing if two triangles are identical. It may be proved by congruence criteria such as

• Side-Side-Side (SSS)
• Side-Angle-Side (SAS)
• Angle-Side-Angle (ASA)
• Right Angle-Hypotenuse-Side(RHS)

These qualities are essential for understanding and solving triangle-related geometry issues.

Formulas of Triangle

Here are the formulae for triangles presented in a table format: