When we think of pyramid we remember the conical shape, where the sides are bounded by triangular surfaces. We see some of the arches are triangular in shape. Triangle is a smallest polygon having three sides. It is a plane figure which is closed. We know that any polygon of more than 3 sides can be split into small triangles. We can form new shapes using triangles of same shapes. In this section let us discuss about the area and perimeter of a triangle, classification of triangles according to their sides and angles, properties of triangles and some problems based on properties of triangles.  

What are Triangles? Triangles are closed figures bounded by three straight lines.

The following figure shows the portion bounded by the three lines.
Triangle Images
The lines, l, m and n intersect at three non-collinear points, A, B and C and form the triangle $\Delta$ ABC.

We can also state that a triangle is formed by joining three non-collinear points in a plane. The line segments joining the non-collinear points are called the sides of the triangle and the three non-collinear points are called the vertices.
Triangle Definition
In the above figure the Triangle formed is $\Delta$ ABC.
The vertices of the triangle are A, B and C
The sides of the $\Delta$ ABC are, AB, BC and CA.
Area of a Traingle = $\frac{1}{2}$ x Base x Height

Perimeter of Triangle = Sum of all the sides
= AB + BC + CA


Triangle Properties:
1. Sum of all the Angles of a triangle is equal to 180o .
In the above triangle $\angle A$ + $\angle B$ + $\angle C$ = 180o

2. The Exterior angle of a triangle is equal to sum of the interior opposite angles.
Triangle Properties
In the above figure $\angle ACD$ = $\angle ABC$ + $\angle BAC$
Triangles are classified in terms of their sides and the angles.

Triangles classified according to their sides are,
1. Scalene Triangle.
2. Isosceles Triangle
3. Equilateral Triangle

Triangles classified according to their angles are,
1. Acute angled triangle.
2. Obtuse angled triangle.
3. Right angled triangle.

A. Triangles classified according to their sides


Scalene Triangle: A triangle in which no sides are equal is called a scalene triangle.
The following figure shows the shape of a scalene triangle.
Classifying Triangles
For a given scalene triangle, we can find the area using the formula

Area = $\frac{1}{2}$ . base $\times$ height
Types of Triangles

The above figure shows the scalene triangle whose base and the height are known.

If the length of the sides of the triangle are known we can find the area using Heron's formula.

The formula can be given by $\sqrt{s(s-a)(s-b)(s-c)}$ where s is the semi perimeter given by s = $\frac{a+b+c}{2}$

For a given scalene triangle if we are given a pair of adjacent sides and the angled included by the two sides are given,
we can find the area using the formula given below.

In the above figure the given adjacent sides are AC and AB and included angle is $\angle CAB$
Area = $\frac{1}{2}$ AB x AC sin $\angle CAB$

If we are given a pair of angles of a triangle and one of the side opposite to the given angle we can find the other sides using
the Law of Sines given by $\frac{a}{sin\angle A}$ = $\frac{b}{sin\angle B}$ = $\frac{c}{sin\angle C}$

We can also solve for remaining sides and the angles of a scalne triangle using Law of cosines given by,
$a^{2}$ = $b^{2}$ + $c^{2}$ - 2 bc cos $\angle A$

$b^{2}$ = $c^{2}$ + $a^{2}$ - 2 ac cos $\angle B$

$c^{2}$ = $a^{2}$ + $b^{2}$ - 2 ab cos $\angle C$


Isosceles Triangle: A triangle whose two of its sides are equal is called an isosceles triangle.
In the following figure, the sides AB = AC.
Isosceles Triangle
Isosceles triangle Theorem: 1 If two sides of a triangle are equal then the angles opposite to them are equal.
In the above triangle, AB = AC => $\angle B$ = $\angle C$

Converse of Isosceles Triangle Theorem: If two angles of a triangle are equal, then the sides opposite to
equal angles of the triangle are equal.
If $\angle B$ = $\angle C$ => AB = AC

Isosceles Triangle Theorem 2: The perpendicular drawn from the vertex of an isosceles triangle bisects the third side.

Isosceles Triangle Theorem 3: The median drawn to the third side of an isosceles triangle is perpendicular to the base (third side).

Isosceles Triangle Theorem 4: The angular bisector of the vertex of an isosceles triangle divides the triangle into two congruent
right triangle.

Area of the isosceles triangle: We can find the area of an isosceles traingle as using the formula,
Area = $\frac{1}{2}$ x Base x Height
(or)
Area = $\frac{1}{2}$ BC x AB sin $\angle ABC$


Equilateral Triangle: A triangle whose sides are of equal measures is called an equilateral triangle.
In an equilateral triangle, each angle measure 60o

The following diagram shows an equilateral triangle whose sides are equal to 2a units.
Equilateral Triangle
The area of an equilateral triangle = $\frac{\sqrt{3}}{4}$ a2, where 2a is the length of the side.

B. Triangles classified according to their angles

Types of Triangle by Angles

Acute Triangles (also called Acute angled triangle): A triangle whose each of the angle measure less than 90o (acute angle) is called an Acute Triangle.

In the above $\Delta ABC$, each of the angles A, B and C are acute.

Right Triangles (also called a right angled triangle): A triangle in which one of the angle measure exactly 90o is called a Right Triangle.

Obtuse Triangle (also called obtuse angled triangle): A triangle in which one of the angle measure more than 90o (obtuse angle) is called an obtuse angled triangle.

IT is not possible for a triangle to have more than one obtuse angle.
In the above triangle $\angle PQR$ is obtuse.
An obtuse angled triangle can be scalene or isosceles, but can never be an equilateral triangle

Area of an obtuse angled triangle = $\frac{1}{2}$ x Base x Height

Height of the triangle will be outside the triangle if one of the side containing the obtuse angle is a base.

Solved Examples

Question 1: The angles of a triangle are in the ratio 2 : 5 : 8. Find the angles of the triangle.
Solution:
 
We are given that the angles of a triangle are 2 : 5 : 8
Let us assume that the angles of the triangle are 2x, 5x and 8x
We know that the sum of the angles of a triangle is equal to 180o .
Sum of the angles = 2x + 5x + 8x = 180
=> 15 x = 180

=> x = $\frac{180}{15}$ = 12

Therefore, the angles of the triangle are, 2 (12), 5 (12), 8 (12) = 24o , 60o , 96o
So that the sum of the angles = 24 + 60 + 96 = 180o
 

Question 2: In the adjoining diagram, $\angle A$ = (x + 5)o , $\angle B$ = (2x + 3)o and $\angle BCD$ = (5x - 14)o.
Find the value of x and hence find the measure of (a) $\angle BCD$ (b) $\angle ACB$
Triangle Solved Examples
Solution:
 
We have $\angle A$ = (x + 5)o , $\angle B$ = (2x + 3)o and $\angle BCD$ = (5x - 14)o
Since the Exterior angle of a triangle is equal to the interior opposite angle,

$\angle BCD$ = $\angle B$ + $\angle A$
=> 5x - 14 = x + 5 + 2x + 3
=> 5x - 14 = 3x + 8
=> 5x - 3x = 8 + 14
=> 2x = 22
=> x = $\frac{22}{2}$
= 11

Therefore, $\angle BCD$ = 5x - 14
= 5 (11) - 14
= 55 - 14
$\angle BCD$ = 41 ---------------------- (a)
$\angle BCD$ + $\angle ACB$ = 180o [linear Pair]
=> 41 + $\angle ACB$ = 180
=> $\angle ACB$ = 180 - 41
$\angle ACB$ = 139o ------------------- (b)