Triangles are geometric shapes with three sides and three angles. They can be categorized based on their angles as right, obtuse, or acute triangles and based on their sides as equilateral, isosceles, or scalene triangles. Key properties of triangles include the angle sum property that the interior angles sum to 180 degrees, Pythagorean theorem relating the sides of a right triangle, and congruence rules to determine if two triangles are identical in shape and size. Triangles are fundamental building blocks that are important across many fields including engineering, trigonometry, and studying distant objects.

1. Triangles
Class : IX A
Done By: Prarthana. A Roll No:37

2. 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.

3. 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.

4. 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°.

5. 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.

6. 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

7. Angle Sum Property
of a Triangle
The sum of angles of a triangle add up to 1800.
A
B C
=180 0
Hence proved

8. 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

9. 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

10. Some more theorems
 Angles opposite to equal sides of an isosceles triangle are
equal.
The sides opposite to equal angles are equal
If 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

11. 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.

12. 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.

13. 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.

14. 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.

15. 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

16. 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