Triangles Class : IX A

Done By: Prarthana. A Roll No:37

Introduction

A triangle is one of the basic shapes of geometry which is determined by any three collinear points. A triangle has three sides and three angles.

Based on their angles they can be characterized to three groups:

Right , obtuse and acute

Based on their sides they can be again divided into three: Equilateral, scalene and isosceles.

Right Triangle

A right triangle has one of its interior angles measuring 90°. The side opposite to the right angle is the hypotenuse; it is the longest side of the right triangle. Right triangles obey the Pythagoras theorem. 

Obtuse triangle

A triangle that has one interior angle that measures more than 90° is an obtuse triangle or obtuse-angled triangle. If the greatest side length is c, then a2 + b2 < c2.

Acute Triangle

A triangle that has all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If the greatest side length is c, then a2 + b2 > c2.

Equilateral Triangle

In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.

Isosceles Triangle

In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length. Some mathematicians define an isosceles triangle as one with at least two equal sides.

Scalene Triangle

In a scalene triangle, all sides are unequal,  and equivalently all angles are unequal. Right triangles are scalene if and only if not isosceles.

Naughty Facts The sum of all the internal angles of a

triangle is always 180o no matter how the triangle is constructed. The length of any side of a triangle is shorter

than the sum of the other two sides.  A triangle can always be split into two right

triangles no matter how the triangle is constructed. An exterior angle of a triangle is an angle that

is a linear pair to an interior angle The sum of the measures of the three exterior

angles (one for each vertex) of any triangle is 360 degrees

Angle Sum Property of a Triangle

The sum of angles of a triangle add up to 180 0.

A

B C

=180 0

Hence proved

Pythagoras Theorem The square of the hypotenuse is equal to the sum of the square of the other two sides.

A

Perpendicular Hypotenuse B C

Base (AC)

2= (AB)2 + (BC) 2

(Hypotenuse) 2 = (Perpendicular) 2 + (Base) 2

Inequality theorem The sum of any two sides of a triangle is greater than the third side.

AB+BC >AC

A

A B + B C

= A B C

C B

And , AC = A C

This shoes that sum of two sides is greater than the third side.

Hence proved

Some more theorems Angles opposite to equal sides of an isosceles triangle are equal.The sides opposite to equal angles are equalIf two sides of a triangle are unequal, the angle opposite to the longer side is larger.In a triangle, the side opposite to the larger angle is the longest.In a triangle, the side opposite to the lesser angle is the shortest

Congruence SAS CONGRUENCE ( SAS)

Two triangles are said to be congruent if two sides and the included angle of a triangle is equal to the two sides and the included angle of another triangle.

SSS congruence ( side side side )

Two triangles are said to be congruent when all sides of a triangle are equal to all sides of another triangle.

ASA congruence ( angle side angle )

Two triangles are said to be congruent when two angles and the included side of a triangle is equal to the two angles and the included side of another triangle.

Aas congruence (angle angle side )

Two triangles are said to be congruent when two angles and a side of a triangle is equal to two angles and a side of another triangle.

Use of triangles Triangles are an important part of geometry. Trigonometry is entirely based on it. Engineering is completely dependent on use of triangles. They are building elements of many structures. Use of congruence makes it possible to study the far away locate objects by assuming them similar to a figure

Conclusion From what I studied I’m able to conclude that triangles are an important part of human lives. Geometry are incomplete without triangles. Triangles have gave rise to things which we make use of in our day to day lives without even noticing them

Thank you