Triangle Basics

What is a Triangle?

A triangle is a plane and closed geometrical figure having three sides, i.e. it is bounded by three line segments.

So, a triangle has:

• three sides
• three angles
• three vertices

Basic Properties of a Triangle

There are some very basic properties of a triangle, that you must be aware of.

• The sum of the three angles of a triangle will always be 180°.

• Sum of any two sides of a triangle will always be greater than the third side.

• Difference between any two sides of a triangle will always be less than the third side.

Types of triangles (According to side)

Equilateral Triangle

It is a triangle which has three equal sides. In the above figure, AB = BC = CA

As all the sides of an equilateral triangle are equal, all of its angles are equal too (each angle is equal to 60°). That is, ∠ABC = ∠BCA = ∠CAB

Isosceles Triangle

It is a triangle which has any of the two sides equal. The third side is of a different length. In the above figure, BC = CA

As two sides of an isosceles triangle are equal, two of its angles are equal too. That is, ∠ABC = ∠CAB

Scalene Triangle

It is a triangle having all unequal sides. That is, no two sides have the same length. As all the sides of a scalene triangle are unequal, all of its three angles are unequal too.

Types of Triangles (According to angle)

Acute-angled Triangle

All the three angles of an acute-angled triangle are acute angles, i.e. less than 90°.

In acute angled triangle:

• Sum of any two angles would be greater than 90°. (Reason: If it’s not, then the third angle will have to be more than 90°. In that case, the triangle will not be an acute-angled triangle.)

• If the lengths of the sides are a, b and c (the largest side being c), then $c^2$ < $a^2$ + $b^2$.

Right-angled Triangle

In a right angled triangle, one of the angles is a right angle, i.e. exactly 90°.

In right angled triangle:

• Sum of two other angles (apart from the right angle) would be equal to 90°. (Reason: We know that the sum of the angles of a triangle is 180°. If one angle is 90°, then the sum of the other two must be 90° too.)
  
  In other words, we can also say that if you ever find that in a triangle, the sum of two angles is equal to the third angle, then it means that it is a right-angled triangle.

• If the lengths of the sides are a, b and c (the largest side being c, also called hypotenuse), then $c^2$ = $a^2$ + $b^2$.

Obtuse-angled Triangle

In an obtuse angled triangle, one of the angles is an obtuse angle, i.e. more than 90°.

In obtuse-angled triangle:

• Sum of two other angles (apart from the obtuse angle) would be less than 90°. (Reason: If it’s not, then the third angle will have to be less than 90°. In that case, the triangle will not be an obtuse-angled triangle.)

• If the lengths of the sides are a, b and c (the largest side being c), then $c^2$ > $a^2$ + $b^2$.

Some more advanced properties of a triangle

Property 1

If a side of triangle is produced (say AB), then the exterior angle so formed (∠CBD) will be equal to the sum of the two interior opposite angles. That is, ∠CBD = ∠BCA + ∠CAB = x° + y°

Property 2

In an equilateral triangle, sum of perpendicular distances of all the three sides from any point inside of triangle is equal to height of the triangle. Let us take a point D inside the triangle, and from it drop perpendiculars on the sides. In the above figure, the height of equilateral triangle ABC, h = DP + DQ + DR