The Triangle and its Properties

A triangle is a closed figure with three sides, three angles, and three vertices. Understanding its properties is fundamental to geometry.

Classification by Sides:
Scalene triangle: all three sides are different lengths
Isosceles triangle: two sides are equal
Equilateral triangle: all three sides are equal

Classification by Angles:
Acute triangle: all angles are less than 90 degrees
Right triangle: one angle is exactly 90 degrees
Obtuse triangle: one angle is more than 90 degrees

Angle Sum Property:
The sum of the three interior angles of any triangle = 180 degrees. This is the most important property of triangles.

Exterior Angle Property:
An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles. If angle ACD is an exterior angle at vertex C, then angle ACD = angle A + angle B.

Sides and Angles Relationship:
The side opposite the greatest angle is the longest side.
In an equilateral triangle, all sides are equal and all angles are 60 degrees.
In an isosceles triangle, the two base angles (angles opposite the equal sides) are equal.

Triangle Inequality:
The sum of any two sides of a triangle is always greater than the third side.
a + b > c, b + c > a, a + c > b.

Pythagoras Theorem (for right triangles):
In a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
Hypotenuse² = Base² + Height² (or c² = a² + b²)

Example 1

Two angles of a triangle are 50 and 70 degrees. Find the third.
Third angle = 180 - 50 - 70 = 60 degrees.

Example 2

An exterior angle of a triangle is 110 degrees. One interior angle at that vertex's opposite side is 45 degrees. Find the other non-adjacent interior angle.
Other angle = 110 - 45 = 65 degrees.

Example 3

In an isosceles triangle, the two equal base angles are each 55 degrees. Find the apex angle.
Apex angle = 180 - 55 - 55 = 70 degrees.

Example 4

Can a triangle have sides 3 cm, 4 cm and 8 cm?
Check: 3 + 4 = 7, which is NOT greater than 8. So no, this triangle cannot exist.

Example 5

In a right triangle, the two legs are 6 cm and 8 cm. Find the hypotenuse.
Hypotenuse² = 6² + 8² = 36 + 64 = 100. Hypotenuse = 10 cm.

Example 6

The sides of a triangle are in ratio 1:1:1. What type of triangle is it?
All sides are equal, so it is an equilateral triangle. Each angle = 60 degrees.

Example 7

In triangle PQR, the exterior angle at Q is 125 degrees. Angle P = 65 degrees. Is angle R = 60 degrees?
Exterior angle at Q = Angle P + Angle R. 125 = 65 + Angle R. Angle R = 60 degrees. Yes.

Key Formulas:
Angle sum: A + B + C = 180 degrees
Exterior angle = sum of two remote interior angles
Pythagoras: c² = a² + b²

Common mistakes

Do not apply Pythagoras theorem to triangles that are not right-angled. Also remember the exterior angle property gives the SUM of the two remote interior angles, not one of them.

Summary

Triangles have important properties including the angle sum (180 degrees), exterior angle property, and the triangle inequality. Pythagoras theorem applies specifically to right-angled triangles. Classification by sides and angles helps categorise triangles.

Practice Problems