ANGLE SUM PROPERTY: HOW TO DESCRIBE ITS OF TRIANGLE

INTRODUCTION:

A plane figure formed by three line segments that are non-parallel to each other is called the triangle.(ANGLE SUM PROPERTY)

If A,B,C are three non-collinear points in the plane of the paper, then the figure made up by the three line segments AB, BC and CA is called a triangle with vertices A,B,C. The triangle contains three vertices A,B and C and three non-parallel line segments AB,BC and CA. This triangle is denoted by ΔABC. AB, BC and CA are sides of ΔABC. Three angles are ˂BAC, <ABC and <ACB.

ELEMENTS or PARTS: The three sides are AB, BC, CA and three angles <A, <B, <C of ΔABC are together called the six parts or elements of ΔABC.

INTERIOR and EXTERIOR of TRIANGLE: We observe that all points in the plane ΔABC are divided into following three parts:

INTERIOR: The part made up by all such points P which are enclosed by ΔABC is called the interior of ΔABC.
EXTERIOR: The part made up by all such point Q which are not enclosed by ΔABC is called the exterior of ΔABC.
TRIANGULAR REGION: The interior of ΔABC itself includes P and R is called the triangular region ΔABC.

TYPES OF TRIANGLES

Naming of triangles by considering the lengths of their sides:-
Scalene Triangle:

A triangle whose no two sides are equal is called a scalene triangle. ΔABC is a scalene triangle.

Isosceles Triangle:

A triangle whose two sides are equal is called an isosceles triangle. ΔABC is an isosceles triangle so AB = AC.

Equilateral Triangle:

A triangle whose all sides are equal to one another, is called an equilateral triangle. ΔABC is an equilateral triangle where AB = AC = BC.

Naming triangles by considering the measures of their angles:-
Acute Triangle :

A triangle whose all the angles are acute is called an acute angled triangle or an acute triangle.

ΔABC is an acute-angled triangle where <A, <B and < C are acute angles. Equilateral triangle is an acute-angled triangle because the measure of its each angle is 60°.

Right Triangle:

A triangle whose one angle is a right angle, is called right-angled triangle or a right triangle.

The side opposite to the right triangle is known as hypotenuse and other two sides are called the legs of the triangles.

ΔABC is a right-angled triangle, where <B = 90° remaining two angles are acute.

Obtuse triangle:

A triangle whose one angle is obtuse, is called an obtuse-angled triangle or an obtuse triangle, where <E is an obtuse angle.

ANGLE SUM PROPERTY OF A TRIANGLE

PROPERTY: The sum of the angles of a triangle is two right angles or 180°

PROOF: Let ΔABC be any triangle. Through A, a line XY parallel to the side BC . Since, XY||BC and the and the transversal AB cuts XY and BC at A and B respectively.

<1 = <4 [ as alternate interior angles are equal ]

Similarly , XY ||BC and the transversal AC cuts XY and BC at A and C respectively.

<2 = <5 [ as alternate interior angles are equal]

Also <3 = <3

Adding the angles on the respective sides, we get

<1 + <2 + <3 = <4 + <5 + <3

But, <4 + <5 + <3 = 180° = 2 right angles.

Hence, the sum of the angles of a triangle is two right angles or 180°.

From the above property, we obtain the following useful results:-

A triangle cannot have more than one right angle.

A triangle cannot have more than one obtuse angle i.e if one angle of the triangle is obtuse, then other two are acute.

In a right triangle, the other two angles are always acute and their sum is 90°.