The Triangle and its Properties

1. TRIANGLE
A closed plane figure bounded by three line segments is called a triangle. Let A,B and C be three noncollinear points

Then the figure formed by the three line segments AB,BC, and CA is called a triangle ABC, denoted by $△$ABC .
$△$ABC has
i) Three sides namely AB, BC, and CA
ii) Three angles namely $△$BAC, $△$ABC, $△$BCA to be denoted by $\angle A,\angle B,\angle C$ respectively
The three sides and three angles of a triangle together are called six parts (or) elements of the triangle
In $△$ABC the points A,B, and C are called its vertices
‘A’ is the vertex opposite to the side BC
‘B’ is the vertex opposite to the side CA
‘C’ is the vertex opposite to the side AB
Interior and Exterior of a Triangle
• The part of the plane enclosed by $△$ABC is called the interior of $△$ABC
• The part of the plane not enclosed by $△$ABC is called the exterior of $△$ABC

In the above figure, points D,E, and F lie in the interior of $△$ABC points P,Q and R lie, in the exterior of $△$ABC , while the points ‘X’ and ‘Y’ lie on $△$ABC

Triangular Region
The interior of a triangle together with its boundary is called the triangular region

Exterior and Interior Angles

• On producing the side be of $△$ABC to a point D, the exterior angle formed is $\angle$ACD and its interior opposite angles are $\angle$CAB and $\angle$ABC, while $\angle$BCA is the interior adjacent angle
• On the producing the side of $△$ABC to a point E, the exterior angle formed is $\angle$CBE and its interior adjacent angle $\angle$CBA
• On producing the side CA of $△$ABC to a point F, the exterior angle formed is $\angle$BAF and its interior opposite angles are $\angle$ABC and $\angle$ACB, while $\angle$BAC is the corresponding interior adjacent angle
Note
The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. The three line segments can form a triangle only when the sum of the lengths of two smaller line segments is greater than the length of the longest one.
2. CLASSIFICATION OF TRIANGLES ACCORDING TO SIDES
There are three types of triangles with respect to the sides
Scalene Triangle
If all three sides of a triangle are of different lengths, then the triangles is called a scalene triangle.
In the figure below, ABC is a triangle in which $\mathrm{AB}\ne \mathrm{BC}\ne \mathrm{CA}$

Therefore $△$ABC is a scalene triangle. All the three angles of a scalene triangle are of different measures $\text{ i.e }\mathrm{\angle }\mathrm{A}\ne \mathrm{\angle }\mathrm{B}\ne \mathrm{\angle }\mathrm{C}$

Isosceles Triangles
A triangle having two sides of equal length is called an isosceles triangle

In the above figure, PQR is a triangle in which $\mathrm{PQ}=\mathrm{PR}\ne \mathrm{QR}$
Therefore $△$PQR is an Isosceles triangle.

The angles opposite to the equal sides of an Isosceles triangle are equal. Since PQ = PR;

we have $\angle Q=\angle R$

Equilateral Triangle
A triangle having all the three sides of the same length is called an equilateral triangle.

In the figure below, XYZ is a triangle in which XY = YZ = ZX. Therefore $△$XYZ is an equilateral triangle. Each angle of an equilateral triangle measures 60°.
3. CLASSIFICATION OF TRIANGLE ACCORDING TO ANGLES
There are three types of triangles with respect to their angles.
Acute Angled Triangle
A triangle with all the three angles acute is called an acute angled triangle

In the figure above, ABC is a triangle in which $\mathrm{\angle }\mathrm{A}={60}^{\circ }\mathrm{\angle }\mathrm{B}={45}^{\circ }\text{ and }\mathrm{\angle }\mathrm{C}={75}^{\circ }$.

Therefore $△$ABC is an acute angled triangle

Obtuse Angled Triangle
A triangle having one obtuse angle is called an obtuse angled triangle

$△$PQR is an obtuse – angled triangle.
In an obtuse – angled triangle one of the angles is obtuse and each of the remaining two angles is acute.
Right Angled Triangle
A triangle is called right angled if the measure of one of its angle is ${90}^0$

The side opposite to the right angle is called the hypotenuse of the triangle .In the above figures $△$LMN is right angled at ‘M’ clearly , LN is the hypotenuse of $△$LMN.
In a right angle triangle one angle measures 90° and each one of the remaining two angles is acute.
4. IMPORTANT TERMS RELATED TO A TRIANGLES
Medians
A line segment joining a vertex of a triangle to the middle point of the side opposite to the vertex is called median. All the three medians of a triangle always pass through a point.
Centroid of a Triangle
The point of intersection of all the three medians of a triangle is called its centroid.

In the given figure, D,E and F are the midpoints of the sides BC, CA and AB respectively of $△$ABC. so AD, BE and CF are the medians of $△$ABC intersecting at a points ‘G’ .

Therefore ‘’G is the centroid of $△$ABC
Altitudes
A line witch passes through a vertex of a triangle, and is a right angles to the opposite side. A triangle has three altitudes, all the three altitudes of a triangle always pass through a point.
Orthocentre of a Triangle
The point of intersection of all the altitudes of a triangle is called its orthocentre

In the given figure; $\mathrm{AL}\perp \mathrm{BC},\mathrm{BM}\perp \mathrm{CA}\text{ and }\mathrm{CN}\perp \mathrm{AB}$ so AL, BM and CN are the altitudes of $△$ABC

Circumcentre of a Triangle
The perpendicular bisectors of the sides of a triangles always intersects at a point

The points of intersection of all the three perpendicular bisectors of the sides of a triangle is called the circumcentre of the triangle. Then circle passing through the three vertices of a triangle is called the circumcircle. In the given figure, the perpendicular bisectors of the sides BC, CA and AB of $△$ABC intersect at a point ‘O’. ‘O’ is the circumcentre of $△$ABC . With ‘O’ as centre and radius OA, we draw a circle passing through three vertices A,B,C of $△$ABC and hence this circle is the circumcircle of $△$ABC .

Incentre of a Triangle
The angle bisectors of the triangle always intersects at a point. The point of intersection of all the angle bisectors of the triangle is called its incentre. The circle touching the sides of a triangle is called the incircle of the triangle.

In the above figure, the angle bisectors of its angles intersect at the points I. Therefore ‘I’ is the incentre of $△$ABC . From ‘I’ draw IL $\perp$BC, with ‘I’ as centre and radius IL, draw a circle, touching the sides of $△$ABC . Hence this circle is incircle of $△$ABC

5. ANGLE SUM PROPERTY OF A TRIANGLE
The sum of the angles of triangle is ${180}^0$
Proof:
Consider a triangle ABC.

Let line XY be parallel to side BC at A.
AB is a transversal that cuts the line XY and AB, at A and B, respectively.
As the alternate interior angles are equal, $\mathrm{\angle }1=\mathrm{\angle }4\text{ and }\mathrm{\angle }2=\mathrm{\angle }5.$

$\mathrm{\angle }4,\mathrm{\angle }3\text{ and }\mathrm{\angle }5$ form linear angles, and their sum is equal to ${180}^0$

$\begin{array}{l}\mathrm{\angle }4+\mathrm{\angle }3+\mathrm{\angle }5={180}^{\circ }\\ \mathrm{\angle }1+\mathrm{\angle }2+\mathrm{\angle }3={180}^{\circ }\end{array}$

Hence, the sum of the three angles of a triangle is ${180}^0$.

6. EXTERIOR ANGLE PROPERTY OF A TRIANGLE
When a side of a triangle is produced, then the exterior angle so formed is equal to the sum of its interior opposite angles.

e.g. If $\angle$ 4 is an exterior angle of $△$ABC, $\angle$1 and $\angle$2 are the interior opposite angles, then $\angle$4 = $\angle$1 + $\angle$2.
The sum of the lengths of any two sides of a triangle is greater than the third side.

7. PYTHAGOREAN THEOREM
If b and c are legs and a is the hypotenuse of a right angled triangle then, $a^2=b^2+c^2$.

Converse of Pythagorean theorem
If the sum of the squares on two sides of a triangle is equal to the square of the third side, then the triangle must be a right-angled triangle.