The document discusses various properties of triangles, including types like isosceles triangles, the angle sum property, triangle inequality, and the Pythagorean theorem. It explains essential concepts such as medians, centroids, altitudes, orthocenters, and bisectors in relation to triangles. Additionally, it describes the relationships between the sides and angles of triangles, including the characteristics of right-angled triangles and Pythagorean triplets.

1. TRIANGLE
AND ITS
PROPERTIES
BY – ARPIT KUMAR JENA

2. INTRODUCTION
• A triangle has many interesting properties.
• In class 6, we have learned some of the
properties of a triangle.
• In this chapter, we shall learn about some more
properties.

3. ISOSCELES TRIANGLE
• A triangle whose two sides are equal in
length is called an isosceles triangle.
• In an isosceles triangle the side of the two
sides known as base.
• The two angles which include the base of the
triangles are called its base angles.
• The angle opposite to the base of an
isosceles triangle is called its vertical angle.

4. PROPERTIES OF AN ISOSCELES
TRIANGLE
• In a triangle, if two sides are equal, then the angles opposite to these
sides are also equal.
• If in a triangle, two angles are equal, then the sides opposite to these
angles are equal in length.

5. EXRERIOR ANGLES OF A
TRIANGLE
In this figure, we can see an exterior angle of
a triangle.
∠A and B are called the interior opposite
∠
angles corresponding to exterior
angle ACD.
∠
∠ACB is the interior adjacent angle at vertex
C.
∠ACD is the exterior angle of ΔABC at point
C.

6. RELATION BETWEEN EXTERIOR ANGLE
AND INTERIOR OPPOSITE ANGLE
• In a triangle, the exterior angle is equal to the sum of two interior opposite
angle.
• Let us assume, ΔXYZ with an exterior angle XZA which lies on the vertex
∠
Z. X and Y are the interior opposite angles.
∠ ∠
• ∠X = 50°
∠Y = 60°
∠XZA = 110°
• Because,
50°+60° = 110°

7. ANGLE SUM
PROPERTY
• Sum of all the three angles of a
triangle is always 180°.
• In this triangle,
30°+80°+70° = 180°
• In an equilateral triangle, all the
sides are equal and each angle
is 60°.
• In an isosceles triangle, two
sides are equal as well as the
opposite angles to them are
equal.

8. TRIANGLE
INEQUALITY
PROPERTY
• The sum of the length of any two
sides of a triangle is greater than
the length of the third side.
• Let sides AB be c
BC be a
CA be b
• Here, c+a>b
a+b>c
b+c>a

9. PYTHAGORAS THEOREM
• Pythagoras, a Greek philosopher, off 6 century B.C., is said to
have found a very important and useful property of right
angled triangle.
• This property is hence named after him.
• However, prior to Pythagoras, around 800 BC, Indian
mathematician Baudhaya had stated and proved this
property of right angled triangle.
• The side opposite to right angle of a right angled triangle, is
called hypotenuse.
• The other two sides are called legs.
• The sides are called base and perpendicular specifically.
• Hypotenuse2
= perpendicular2
+ base2

10. CONVERSE OF PYTHAGORAS
THEOREM
• If the square of one side of a triangle is equal to the sum of the
squares of the other two sides, then it is a right angled triangle with
the angle opposite to the first side as a right angle.

11. PYTHAGOREAN TRIPLET
• Three positive integers, a, b and c are said to form a Pythagorean triplet
if,
c2
= a2
+b2
• For two positive integers m and n, n > m
Then Pythagorean triplet is
a = (n2
– m2
) b = (2 mn) c = (n2
+ m2
)
• Examples of Pythagorean triplet:- (3, 4, 5); (5, 12, 13); (7, 24, 25) are
Pythagorean Triplets obtained by taking n = 2, m = 1; n = 3, m = 2; n =
4, m = 3 respectively.

12. MEDIANS AND CENTROID OF A
TRIANGLE
• A median is the line segment that joins a vertex to
the mid-point of the opposite side.
• A triangle has three medians.
• The three medians always meet at a single point
and this point is called centroid.
• The medians of a equilateral triangle are equal.
• The medians to the sides of an isosceles triangle
are equal.
• The medians to the base of an isosceles
triangle is perpendicular to the base.
• The centroid of a triangle divides each one of the
medians in the ratio of 2 : 1.

13. ALTITUDE AND ORTHOCENRE
OF A TRIANGLE
• An altitude of a triangle is the line segment from a vertex of the
triangle, perpendicular to the opposite side.
• Every triangle has three altitudes, one from each vertex.
• The three altitudes always meet at the single point and this point is
called orthocentre.
• The altitudes of an equilateral triangle are equal.
• The altitudes drawn on equal sides of an isosceles triangle are equal.
• The ortho centre of an acute angled triangle lies in the interior of the
triangle.
• The orthocentre of a right triangle is the vertex at the right angle.
• The orthocentre of an obtuse angled triangle lies in the exterior of the
triangle.

14. PERPENDICULAR BISECTOR OF
SIDES OF A TRIANGLE
• The perpendicular bisector of his side of the
triangle is the line that bisects the side and is
perpendicular to it.
• There can be 3 perpendicular bisectors of sides
of the triangle.
• The point of concurrence of the perpendicular
bisector's of the sides of a triangle is called the
circumcentre of the triangle.
• The circle passing through the vertices of the
triangle is called the circumcircle of the triangle.
• Radius of the circle is called the circumradius.

15. ANGLE BISECTOR OF A TRIANGLE
• An angle bisector of a triangle is a line segment that
bisects an angle of the triangle and has its other and
on the side opposite to that angle.
• The point of concurrence of the angle bisectors of a
triangle is called the incentre of the triangle.
• The incentre of her triangle is equidistant from its
sides.
• The incircle of a triangle is the circle that just
touches the sides of the triangle.
• The centre of the incircle is the incentre of a triangle.
• The radius of the incircle of a triangle is called its
inradius.