Triangles can be classified based on side lengths as equilateral, isosceles, or scalene triangles and based on angles as acute, obtuse, or right triangles. The three main properties of triangles are the angle sum property, exterior angle property, and Pythagorean theorem. Secondary parts of a triangle include the median, altitude, perpendicular bisector, and angle bisector. Triangles can be proven congruent using the SSS, SAS, ASA, AAS, or RHS criteria. Inequalities in triangles relate longer sides to larger angles and shorter sides to smaller angles. Important centers of a triangle include the incenter, circumcenter, centroid, and orthocenter.

9. Triangles1.

10. What are
triangles ?? ??1.

11. Types of
Triangles
2

12. On Basis of Length of Sides, there are 3 types of Triangles
• Equilateral Triangle
• Isosceles Triangle
• Scalene Triangle
On Basis of Angles, there are 3 types of triangles
• Acute Angled Triangle
• Obtuse Angled Triangle
• Right Angled Triangle
TYPES OF TRIANGLES

13. Properties
A Triangle.
3

14. Properties
Pythagorus
theoram
Exterior
angle
property
Angle sum
property

15. • Angle sum property-
Angle sum Property of a Triangle is that the sum of all interior angles of
a Triangle is equal to 180˚.
• Exterior angle property-
Exterior angle Property of a Triangle is that An exterior angle of the
Triangle is equal to sum of two opposite interior angles of the Triangle
• Pythagorus theoram-
Pythagoras Theorem is a theorem given by Pythagoras. The theorem is
that In a Right Angled Triangle the square of the hypotenuse is equal to
the sum of squares of the rest of the two sides.

16. Properties of an isosceles triangle
• Angle opposite to the equal sides of an
isosceles triangle are equal.
• The sides opposite to the equal angles of
a triangle are equal.

17. Secondary
Parts
Triangle.
4

18. 1. The Line Segment joining the midpoint of the base of the Triangle is
called Median of the Triangle.
OR
2. A Line Segment which connects a vertex of a Triangle to the
midpoint of the opposite side is called Median of the Triangle.
MEDIAN

19. Altitudes of a triangle
The Line Segment drawn from a Vertex of a
Triangle perpendicular to its opposite side is
called an Altitude or Height of a Triangle.
ALTITUDE

20. Perpendicular bisector
A line that passes through midpoint of the
triangle
or the line which bisects the third side of the
triangle and is perpendicular to it is called the
Perpendicular Bisector of that Triangle.
Perpendicular
Bisector

21. Angle Bisector
A line segment that bisects an angle of a triangle
Is called Angle Bisector of the triangle.
ANGLE
BISECTOR

22. Congruency
OF A
Triangles
5

23. •Two figures are congruent, if they are
of the same shape and of the same
size.
•Two circles of the same radii are
congruent.
•Two squares of the same sides are
congruent.

24. SSS criteria of congruency
If the three sides of one Triangle
are equal to the three sides of
another Triangle. Then the
triangles are congruent by the
SSS criteria.
SSS criteria is called Side-
Side-Side criteria of congruency.

25. SAS criteria of congruency
If two sides and the angle included
between them is equal to the
corresponding two sides and the
angle between them of another
triangle. Then the both triangles are
congruent by SAS criteria i.e. Side-
Angle-Side Criteria of Congruency.

26. ASA criteria of congruency
If two angles and a side of a Triangle is
equal to the corresponding two angles
and a side of the another triangle then
the triangles are congruent by the ASA
Criteria i.e. Angle Side-Angle Criteria of
Congruency.

27. AAS criteria of congruency
If two angles and one side of one
triangle are equal to angles to two
angles and the corresponding side of
the other triangle then the two triangles
are congruent

28. RHS criteria of congruency
If the hypotenuse, and a leg of one right
angled triangle is equal to corresponding
hypotenuse and the leg of another right
angled triangle then the both triangles are
congruent by the RHS criteria i.e. Right
Angle-Hypotenuse-Side Criteria of
Congruency.

29. Inequalities
A Triangle.
6

30. • In a triangle ,angle opposite to the longer
side is larger.
• In a triangle, side opposite to the
larger(greater) angle is longer.
• Sum of any two sides of a triangle is
greater than the third side.
• Difference of any two sides of a triangle
is smaller than the third side.

31. Some
Facts About
Triangles.
7

32. Centres of a circle
• Incentre-
The three angle bisectors of a triangle meet in
one point called the incentre. It is the centre
of the incircle, the circle inscribed by the
triangle

33. • Circumcentre-
Three perpendicular bisectors of the sides of
the triangle meet in one point called
circumcentre. It is the centre of the
circumcircle, the circle circumscribed about
the circle.

34. • Centroid-
The three medians meet in the centroid of the
centre or center of the mass(centre of
gravity).

35. • Orthocentre-
The three altitudes of a tiangle meet in one
point called the orthocentre.

36. PYTHAGORAS EUCLID PASCAL
MATHEMATICIANS RELATED TO TRIANGLES

37. SUBMITTED TO:
MRS.RAMAN KAUR
SUBMITTED BY:
ARYAN BAWA
CLASS: 9TH ‘A’