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Revision Notes on Triangles

Triangle

A closed figure with three sides is called a Triangle. It has three vertex, sides and Angles.

Triangle

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 Scalene
Isosceles Length of two sides are equal Isosceles
Equilateral Length of all three sides are equal Equilateral

2. There are three types of triangles on the basis of angles.

Name of Triangle Property Image
Acute All the three angles are less than 90° Acute
Obtuse One angle is greater than 90° Obtuse
Right One angle is equal to 90° Right

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.

Two circles are congruent if their radii are same.

2. Two squares are congruent if their sides are equal.

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 “≅”.

Congruence of Triangles

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 (Side-Angle-Side) Congruence rule

If the two sides and the including angle of one triangle is equal to another triangle then they are called congruent triangles.

 SAS Congruence rule
2.

ASA (Angle-Side-Angle) Congruence rule

If the two angles and the including side of one triangle is equal to another triangle then they are called congruent triangles.

 ASA Congruence rule
3.

AAS (Angle-Angle-Side) 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.

 AAS Congruence rule
4.

SSS (Side-Side-Side) 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.

 SSS Congruence rule
5.

RHS (Right angle-Hypotenuse-Side) Congruence rule

If there are two right-angled triangles then they will be congruent if their hypotenuse and any one side are equal. 

 RHS Congruence rule

Remark

1. SSA and ASS do not show the congruency of triangles.

2. 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.

∆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.

Isosceles Triangle.

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

Inequalities in a TriangleTheorem 1: In a given triangle if two sides are unequal then the angle opposite to the longer side will be larger.

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.

Triangle

Example

Show whether the inequality theorem is applicable to this triangle or not?

Triangle

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.


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