Triangle Part -1 ( Basic)

Triangle

A triangle is a closed plane figure which has three sides. The word "triangle" is a Greek word. "Tri" means "three". Hence, it refers to a shape consisting of three internal angles and three sides.

The general shape of a triangle is shown below.

A triangle is named by the letters of the vertices. Normally, upper case letters are used to denote the vertices. The triangle shown above is called △ ABC

Triangle Definition

A triangle is a polygon having three sides. A perpendicular drawen from a vertex to the opposite side is called as the height of triangle and is denoted as 'h'.

Triangle Angles

A triangle is a closed plane figure which has three angles. The sum of all the interior angles of a triangle is 180 degrees. The interior angles are denoted by the name of the vertex.

∠A + ∠B + ∠C = 180^o.

Classifying Triangles

Triangles are classified on the basis of their angles and sides:

Types of triangles based on their sides

1. Scalene triangle

 A scalene triangle is a triangle that has no equal sides. The following is a scalene triangle.

2. Isosceles triangle

An isosceles triangle is a triangle that has two equal sides. The following is an isosceles triangle.

3. Equilateral triangle

An equilateral triangle is a triangle that has three equal sides. The following is an equilateral triangle.

Triangles based on their angles

1. Right angle triangle

A right triangle has a 90 degrees angle.The following is a right triangle.

2. Obtuse angle triangle

An obtuse triangle has one angle that is bigger than 90 degrees (Obtuse angle). The following is an obtuse triangle. 

3. Acute angle triangle

In an acute triangle, all angle are less than 90 degrees, so all angles are acute angles.The following is an acute triangle.

Properties of a triangle

Property 1: Triangle Sum Theorem - The sum of the 3 angles in a triangle is always 180°.

Example :

Property 2: The sum of an interior angle and its adjacent exterior angle is 180⁰.

Example:

Property 3 : Exterior Angle Theorem : An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Example:

Here are some facts to remember:

• The three angles in a triangle always add up to 180⁰
• The three angles of an equilateral triangle are all equal to 60⁰
• Two angles of an isosceles triangle are equal.
• One angle of a right-angled triangle is 90⁰
• All angles of an acute-angled triangle are acute angles, thus smaller than 90⁰
• One angle of an obtuse-angled triangle is obtuse, thus larger than 90⁰ and smaller than 180⁰

• The six elements of a triangle are its three angles and the three sides.
• An exterior angle of a triangle is formed when a side of a triangle is produced. At each vertex, you have two ways of forming an exterior angle.
• A property of exterior angles: The measure of any exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles.
• The angle sum property of a triangle: The total measure of the three angles of a triangle is 180°.
• Property of the lengths of sides of a triangle: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. The difference between the lengths of any two sides is smaller than the length of the third side.

Examples on above properties: -

Example:1

Find the value of x in the following triangle.

Solution:

x + 24° + 32° = 180° (sum of angles is 180°) 

x + 56° = 180°

x = 180° – 56° = 124°

Example :2

Find the values of x and y in the following triangle.

Solution:

x + 50° = 92° (sum of opposite interior angles = exterior angle) 

x = 92° – 50° = 42°

y + 92° = 180° (interior angle + adjacent exterior angle = 180°.) 

y = 180° – 92° = 88°