What is a Triangle?
A triangle is a polygon with three sides, three vertices, and three angles. In everyday life, you might have seen objects that are triangular in shape, such as sandwiches, pyramids, cloth hangers, and so on.
It is one of the most basic shapes in geometry. A triangle is represented by the symbol \(\triangle\). For example, the triangle given below is represented as \(\triangle\)ABC, where A, B and C are the vertices of the triangle, whereas a, b and c represent the side measure of the triangle.
The Properties of a Triangle
The properties of a triangle are defined on the basis of the three of its sides and three angles. Let us explore these properties by taking \(\triangle\)ABC as the reference triangle.
1. The sum of all the angles of a triangle is \(180^{\circ}\).
\(\triangle\)ABC has three interior angles named ∠A, ∠B and ∠C. So, as per the property:
∠A + ∠B + ∠C = \(180^{\circ}\)
2. The sum of the lengths of any two sides is greater than the length of the third side.
\(\triangle\)ABC has three sides named a, b and c. The side opposite to ∠A measures a units, the side opposite to ∠B measures b units, and similarly, the side opposite to ∠C measures c units. So, as per the property:
(a + b) > c , (b + c) > a , (c + a) > b
3. The difference between the length of any two sides is less than the length of the third side.
So, as per the property:
(a – b) < c , (b – c) < a , (c – a) < b
4. The measure of the angle opposite to the longer side is larger, and the angle opposite to the smaller side is smaller.
So, as per the property:
If the order of the length of the sides is a > b > c,
the order of the measure of the angles will be ∠A > ∠B > ∠C
The converse of the above property is:
The side opposite the larger angle is the largest in length, and the side opposite the smaller angle is the smallest in length.
If the order of the angles is ∠A > ∠B > ∠C,
the the order of the sides will be a > b > c
5. Exterior angle property
The measure of the exterior angle of a triangle is equal to the sum of its interior opposite angles.
Exterior angle, ∠CBD = ∠A + ∠C
Where,
∠A and ∠C are the interior angles.
6. If any two angles of a triangle are equal to two angles of another triangle, their third angles are also equal and the two triangles are called similar triangles.
This property can also be stated as: if the corresponding angles of two triangles are equal, the triangles are called similar triangles.
Let us take two triangles. \(\triangle\)ABC and \(\triangle\)DEF are given such that
∠A = ∠D and ∠B = ∠E, then ∠C must be equal to ∠F
Therefore, \(\triangle\)ABC and \(\triangle\)DEF are similar, that is, \(\triangle\)ABC ~ \(\triangle\)DEF.
7. The area of a triangle is half the product of its base and height.
Area of a triangle, A = \(\frac{1}{2}\) × Base × Height
A = \(\frac{1}{2}\) x b x h
Where b is the base and h is the height of the triangle.
8. The perimeter of a triangle, P = sum of all its three sides
P = a + b + c
Solved Examples
Example1: In a \(\triangle\)ABC, the measure of ∠A = \(60^{\circ}\) and ∠B = \(60^{\circ}\) . Find the measure of ∠C.
Solution:
∠A + ∠B + ∠C = \(180^{\circ}\) [The sum of all angles of a triangle is \(180^{\circ}\) ]
\(60^{\circ}\) + \(60^{\circ}\) + ∠C = \(180^{\circ}\) [Substitute values]
∠C = \(180^{\circ}\) – \(120^{\circ}\) [Subtract 120 from both sides]
∠C = \(60^{\circ}\)
Hence, the measure of ∠C is \(60^{\circ}\) .
Example 2: For the \(\triangle\)ABC given below, arrange the interior angles in increasing order of their measure.
Solution:
In the given \(\triangle\)ABC, the measure of the sides are: a = 10 cm, b = 8 cm, and c = 6 cm.
The decreasing order of side lengths is 10 > 8 > 6, that is, a > b > c.
In a triangle, the measure of the angle opposite to the longer side is larger, or the angle opposite to the smaller side is smaller.
Hence, the measure of angles in increasing order will be ∠A > ∠B > ∠C.
Example 3:
Find the area of the triangle whose base is 8 inches long and height is 12 inches.
Solution:
Area of triangle = \(\frac{1}{2}\) × b× h [Area of a triangle formula]
= \(\frac{1}{2}\) (8) × (12) [Substitute values of b and h]
= \(\frac{1}{2}\) (96) [Multiply]
= 48 [Simplify]
Hence, the area of the triangle is 48 \(in^2\)
Example 4: In the triangle given below, find the value of the exterior angle x.
Solution:
In the triangle, ∠x is opposite the interior angles ∠64 and ∠45.
∠x = \(64^{\circ}\) + \(45^{\circ}\) [Exterior angle property]
∠x = \(109^{\circ}\) [Add]
Hence, ∠x is equal to \(109^{\circ}\).
An equiangular triangle is a type of triangle having three equal interior angles. Each of the interior angles of this triangle measures 60 degrees.
Acute triangle:
When all three angles of a triangle measure less than 90 degrees, it is known as an acute triangle.
Obtuse triangle:
When any one angle of a triangle is greater than 90 degrees, it is known as an obtuse triangle.
Right triangle:
When one angle of a triangle measures 90 degrees, it is known as a right triangle.
In an isosceles triangle, two sides are equal in length. For such a triangle, two angles opposite to the equal sides are equal in measure.
Half of the perimeter of a triangle is known as its semi-perimeter. It is represented by ‘s’.
In an equilateral triangle, all sides are equal, and in an isosceles triangle, any two sides are equal. So, an equilateral triangle satisfies the condition for an isosceles triangle. Hence, an equilateral triangle is also an isosceles triangle.