NCERT Notes for Class 9 Maths Chapter 5 Triangles
Class 9 Chapter 5 Triangles Notes
Chapter Name  Triangles Notes 
Class  CBSE Class 9 
Textbook Name  NCERT Mathematics Class 9 
Related Readings 

Triangle
A closed figure with three sides is called a Triangle. It has three vertex, sides and Angles.
Types of Triangle
1. There are three types of triangles on the basis of the length of the sides.
Name of Triangle 
Property  Image 
Scalene 
Length of all sides are different  
Isosceles 
Length of two sides are equal  
Equilateral 
Length of all three sides are equal 
Name of Triangle 
Property 
Image 
Acute 
All the three angles are less than 90°  
Obtuse 
One angle is greater than 90° 

Right 
One angle is equal to 90° 
Congruence
If the shape and size of two figures are same then these are called Congruent.
1. Two circles are congruent if their radii are same.2. Two squares are congruent if their sides are equal.
Congruence of Triangles
A triangle will be congruent if its corresponding sides and angles are equal.
The symbol of congruent is “≅”.
AB = DE, BC = EF, AC = DF
m∠A = m∠D, m∠B = m∠E, m∠C = m∠F
Here, ∆ABC ≅ ∆DEF
Criteria for Congruence of Triangles
S.No. 
Rule 
Meaning 
Figure 
1. 
SAS (SideAngleSide) Congruence rule 
If the two sides and the including angle of one triangle is equal to another triangle then they are called congruent triangles. 

2. 
ASA (AngleSideAngle) Congruence rule 
If the two angles and the including side of one triangle is equal to another triangle then they are called congruent triangles. 

3. 
AAS (AngleAngleSide) Congruence rule 
If any two pairs of angles and a pair of the corresponding side is equal in two triangles then these are called congruent triangles. 

4. 
SSS (SideSideSide) Congruence rule 
If all the three sides of a triangle are equal with the three corresponding sides of another triangle then these are called congruent triangles. 

5. 
RHS (Right angleHypotenuseSide) Congruence rule 
If there are two rightangled triangles then they will be congruent if their hypotenuse and any one side are equal. 
Remark
 SSA and ASS do not show the congruency of triangles.
 AAA is also not the right condition to prove that the triangles are congruent.
Example: Find the ∠P, ∠R, ∠N and ∠M if ∆LMN ≅ ∆PQR.
Solution
If ∆ LMN ≅ ∆PQR, then
∠L = ∠P
∠M = ∠Q
∠N = ∠R
So,
∠L = ∠P = 105°
∠M = ∠Q = 45°
∠M + ∠N + ∠L = 180° (Sum of three angles of a triangle is 180°)
45° + 105° + ∠N = 180°
∠N = 180° 45° + 105°
∠N = 30°
∠N = ∠R = 30°
Some Properties of a Triangle
If a triangle has two equal sides then it is called an Isosceles Triangle.
1. Two angles opposite to the two equal sides of an isosceles triangle are also equal.2. Two sides opposite to the equal angles of the isosceles triangle are also equal. This is the converse of the above theorem.
Inequalities in a Triangle
a > b, if and only if ∠A > ∠B
Longer sides correspond to larger angles.
Theorem 2: In the given triangle, the side opposite to the larger angle will always be longer. This is the converse of above theorem.
Theorem 3: The sum of any two sides of a triangle will always be greater than the third side.
Example: Show whether the inequality theorem is applicable to this triangle or not?
Solution
The three sides are given as 7, 8 and 9.
According to inequality theorem, the sum of any two sides of a triangle will always be greater than the third side.
Let’s check it
7 + 8 > 9
8 + 9 > 7
9 + 7 > 8
This shows that this theorem is applicable to all the triangles irrespective of the type of triangle.