Triangles

This chapter plays a significant role in various aspects of our daily lives both in practical applications and in the understanding of mathematical and geometric concepts. Triangles are among the most fundamental and versatile shapes in the world of geometry. These three-sided wonders pop up in various aspects of our daily lives from architecture to art and engineering to navigation. Their unique properties and applications make triangles an essential element of our world.

Triangles

A triangle is a polygon defined by its three sides, three vertices and three angles. The vertices of this triangle are denoted as A, B and C.

Angle Sum Property of a Triangle

The Angle Sum Property of a Triangle states that the combined measure of the angles inside a triangle always amounts to either 180° or is equivalent to the measure of two right angles.
In a triangle defined by its vertices A, B and C, this property can be mathematically expressed as follows:

Exterior Angle property

The exterior angle of a triangle is equal in measurement to the combined measure of the two interior angles opposite to it.

Exterior angle = Sum of two opposite interior angles

Classification of Triangles on the Basis of their Sides

Classifications based on the length of their sides include three main types:

Classification of Triangles on the Basis of their Angles

Classifications based on their angles include three main types:

Congruent Triangles

Two triangles are said to be congruent when each angle in one triangle matches the corresponding angle in the other and each side in one triangle is equal in length to the corresponding side in the other triangle.

Key Points to Keep in Mind:

→ The total length of any pair of sides in a triangle must be greater than the length of the third side.
→ The difference in lengths between any two sides of a triangle must be less than the length of the third side.
→ The sum of the interior angles within a triangle is always equal to 180°.
→ The measurement of an exterior angle in a triangle is equal to the combined measurements of the interior angles opposite to it.

Rules for Congruent Triangles

The rules for congruency of triangles are shown in the table:

Pythagoras’ Theorem

In a right-angled triangle, the square of the longest side (the hypotenuse) is equal to the sum of the squares of the other two sides.