Class 10 Mathematics – Geometry (Part 1)
Chapter: Triangles – Introduction &
Similarity of Triangles
Introduction
A triangle is a closed figure formed by joining three line segments. It has:
3 sides
3 angles
3 vertices
Triangles are one of the most important topics in Class 10 Geometry because they are used in
trigonometry, coordinate geometry, construction, and real-life applications.
Types of Triangles
Based on Sides
1. Equilateral Triangle
All three sides are equal.
All three angles are 60°.
Example:
AB = BC = CA
2. Isosceles Triangle
Two sides are equal.
Two angles are equal.
Example:
AB = AC
,3. Scalene Triangle
All sides are different.
All angles are different.
Based on Angles
1. Acute Triangle
All angles are less than 90°.
2. Right Triangle
One angle is exactly 90°.
3. Obtuse Triangle
One angle is greater than 90°.
Basic Properties of Triangles
Property 1
The sum of the interior angles of a triangle is:
180∘\boxed{180^\circ}180∘
Example:
50° + 60° + 70° = 180°
Property 2
The exterior angle of a triangle equals the sum of the two opposite interior angles.
,Example:
Exterior angle = 120°
Opposite interior angles = 50° and 70°
50° + 70° = 120°
Similar Triangles
Two triangles are similar if:
Their corresponding angles are equal.
Their corresponding sides are proportional.
Symbol:
△ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF
Conditions for Similarity
1. AA (Angle–Angle) Criterion
If two corresponding angles are equal, then the triangles are similar.
Example:
∠A = ∠D
∠B = ∠E
Therefore,
△ABC ∼ △DEF
2. SAS (Side–Angle–Side) Criterion
If:
Two pairs of corresponding sides are proportional, and
The included angle is equal,
, then the triangles are similar.
3. SSS (Side–Side–Side) Criterion
If all three corresponding sides are proportional, then the triangles are similar.
Example:
AB/DE = BC/EF = AC/DF
Therefore,
△ABC ∼ △DEF
Properties of Similar Triangles
If
△ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF
then:
Corresponding angles are equal.
∠A=∠D\angle A=\angle D∠A=∠D ∠B=∠E\angle B=\angle E∠B=∠E ∠C=∠F\angle C=\angle
F∠C=∠F
Corresponding sides are proportional.
ABDE=BCEF=ACDF\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}DEAB=EFBC
=DFAC
Ratio of Areas
The ratio of the areas of two similar triangles is equal to the square of the ratio of their
corresponding sides.
Area of △ABCArea of △DEF=(ABDE)2\boxed{ \frac{\text{Area of }\triangle ABC}
{\text{Area of }\triangle DEF} = \left(\frac{AB}{DE}\right)^2
}Area of △DEFArea of △ABC=(DEAB)2
Chapter: Triangles – Introduction &
Similarity of Triangles
Introduction
A triangle is a closed figure formed by joining three line segments. It has:
3 sides
3 angles
3 vertices
Triangles are one of the most important topics in Class 10 Geometry because they are used in
trigonometry, coordinate geometry, construction, and real-life applications.
Types of Triangles
Based on Sides
1. Equilateral Triangle
All three sides are equal.
All three angles are 60°.
Example:
AB = BC = CA
2. Isosceles Triangle
Two sides are equal.
Two angles are equal.
Example:
AB = AC
,3. Scalene Triangle
All sides are different.
All angles are different.
Based on Angles
1. Acute Triangle
All angles are less than 90°.
2. Right Triangle
One angle is exactly 90°.
3. Obtuse Triangle
One angle is greater than 90°.
Basic Properties of Triangles
Property 1
The sum of the interior angles of a triangle is:
180∘\boxed{180^\circ}180∘
Example:
50° + 60° + 70° = 180°
Property 2
The exterior angle of a triangle equals the sum of the two opposite interior angles.
,Example:
Exterior angle = 120°
Opposite interior angles = 50° and 70°
50° + 70° = 120°
Similar Triangles
Two triangles are similar if:
Their corresponding angles are equal.
Their corresponding sides are proportional.
Symbol:
△ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF
Conditions for Similarity
1. AA (Angle–Angle) Criterion
If two corresponding angles are equal, then the triangles are similar.
Example:
∠A = ∠D
∠B = ∠E
Therefore,
△ABC ∼ △DEF
2. SAS (Side–Angle–Side) Criterion
If:
Two pairs of corresponding sides are proportional, and
The included angle is equal,
, then the triangles are similar.
3. SSS (Side–Side–Side) Criterion
If all three corresponding sides are proportional, then the triangles are similar.
Example:
AB/DE = BC/EF = AC/DF
Therefore,
△ABC ∼ △DEF
Properties of Similar Triangles
If
△ABC∼△DEF\triangle ABC \sim \triangle DEF△ABC∼△DEF
then:
Corresponding angles are equal.
∠A=∠D\angle A=\angle D∠A=∠D ∠B=∠E\angle B=\angle E∠B=∠E ∠C=∠F\angle C=\angle
F∠C=∠F
Corresponding sides are proportional.
ABDE=BCEF=ACDF\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}DEAB=EFBC
=DFAC
Ratio of Areas
The ratio of the areas of two similar triangles is equal to the square of the ratio of their
corresponding sides.
Area of △ABCArea of △DEF=(ABDE)2\boxed{ \frac{\text{Area of }\triangle ABC}
{\text{Area of }\triangle DEF} = \left(\frac{AB}{DE}\right)^2
}Area of △DEFArea of △ABC=(DEAB)2
