A triangle is a closed figure made of three line segments. It has three sides, three vertices and three angles. Triangles are the simplest polygons, and their properties are the gateway to understanding all other polygons. Every triangle is rigid, which is why triangular frames are used in bridges, roofs and cranes. In this chapter we study the angle sum property, the exterior angle property, the special median and altitude, and the all-important Pythagoras theorem.
The triangle is more than a shape; it is a collection of beautiful mathematical relationships. The three interior angles always add to 180 degrees. An exterior angle equals the sum of the two opposite interior angles. The sum of any two sides is always greater than the third side. In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. These properties let us solve for unknown sides and angles, prove statements, and apply geometry to construction and measurement.
A triangle ABC has: - Three sides: AB, BC and CA. - Three vertices: A, B and C. - Three angles: angle A, angle B and angle C. The triangle is written as triangle ABC. The sum of the three sides is called its perimeter.
By sides, a triangle is: - Equilateral: all three sides equal, all angles equal to 60 degrees. - Isosceles: two sides equal, and the angles opposite them equal. - Scalene: all three sides different.
By angles, a triangle is: - Acute angled: all angles less than 90 degrees. - Right angled: one angle exactly 90 degrees. - Obtuse angled: one angle greater than 90 degrees.
The sum of the three interior angles of a triangle is always 180 degrees: angle A + angle B + angle C = 180 degrees This property holds for every triangle. For example, if two angles of a triangle are 50 and 70 degrees, the third angle is 180 - (50 + 70) = 60 degrees. In an equilateral triangle, each angle is 180/3 = 60 degrees.
To find a missing angle, subtract the sum of the known angles from 180. If a triangle has angles 65 and 45, the missing angle is 180 - 110 = 70 degrees. In an isosceles triangle with a vertex angle of 40 degrees, the two equal base angles are each (180 - 40)/2 = 70 degrees.
When one side of a triangle is extended, the angle formed outside the triangle is called an exterior angle. The exterior angle is supplementary to the adjacent interior angle. More importantly, an exterior angle is equal to the sum of the two opposite interior angles: Exterior angle = Sum of the two opposite interior angles For example, if a triangle has interior angles 60 and 70 degrees, the exterior angle at the third vertex is 60 + 70 = 130 degrees. This property is used to find unknown angles quickly.
A median of a triangle is the line segment joining a vertex to the midpoint of the opposite side. Every triangle has three medians, and all three medians pass through a common point called the centroid. A median divides the triangle into two triangles of equal area.
An altitude of a triangle is the perpendicular line segment drawn from a vertex to the opposite side (or its extension). Every triangle has three altitudes, and they meet at a point called the orthocentre. The length of the altitude is used to find the area of the triangle, since area = (1/2) x base x height.
In a right triangle, the two sides forming the right angle are themselves the altitudes to each other, and the third altitude is drawn to the hypotenuse.
The sum of the lengths of any two sides of a triangle is always greater than the length of the third side: AB + BC > CA, BC + CA > AB, CA + AB > BC For example, sides 3, 4 and 5 form a triangle because 3 + 4 > 5. Sides 2, 3 and 8 do not form a triangle because 2 + 3 = 5, which is not greater than 8. This property is used to check whether three given lengths can form a triangle.
In a right-angled triangle, the side opposite the right angle is called the hypotenuse, and it is the longest side. The Pythagoras theorem states: (Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2 For a right triangle with sides a, b and hypotenuse c: c^2 = a^2 + b^2 For example, if a triangle has sides 3 and 4 and the right angle is between them, the hypotenuse is 5 because 5^2 = 3^2 + 4^2 = 9 + 16 = 25. The converse of the theorem is also true: if the square of one side equals the sum of squares of the other two sides, then the triangle is right angled.
If the hypotenuse and one side are known, the other side can be found by: a^2 = c^2 - b^2 For example, if the hypotenuse is 13 and one side is 5, the other side squared is 169 - 25 = 144, so the side is 12.
| Basis | Type | Condition |
|---|---|---|
| Sides | Equilateral | All three sides equal |
| Sides | Isosceles | Two sides equal |
| Sides | Scalene | No two sides equal |
| Angles | Acute angled | All angles less than 90 degrees |
| Angles | Right angled | One angle is 90 degrees |
| Angles | Obtuse angled | One angle more than 90 degrees |
| Property | Statement | Formula/Example |
|---|---|---|
| Angle sum | Sum of interior angles | A + B + C = 180 degrees |
| Exterior angle | Equals sum of opposite interior angles | Exterior = A + B |
| Median | Vertex to midpoint of opposite side | Three medians meet at centroid |
| Altitude | Perpendicular from vertex to opposite side | Area = (1/2) x base x height |
| Triangle inequality | Sum of any two sides > third side | 3 + 4 > 5 |
| Pythagoras theorem | Square of hypotenuse | c^2 = a^2 + b^2 |
The triangle, with its angle sum property, exterior angle property, medians and altitudes, triangle inequality and Pythagoras theorem, is a treasure trove of geometric relationships. These properties allow us to find unknown sides and angles, verify whether triangles can be formed, and solve real-life problems involving right-angled situations like ladders against walls and distances across fields. The Pythagoras theorem in particular connects geometry with algebra and is among the most used results in all of mathematics. A thorough understanding of triangles now will make the study of congruence, perimeter and area, and solid shapes much easier.