📘 Triangles Class 10 Notes (Chapter 6)

🔷 Introduction

Triangles are one of the most important topics in geometry. In simple words, a triangle is a closed figure made by joining three line segments.

Since a triangle has three sides, three angles, and three vertices, it forms the foundation of many geometry concepts.

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🔹 What is a Triangle?

A triangle is a polygon with three sides.

For example:
△ABC

Here:

• AB, BC, and AC are sides
• ∠A, ∠B, and ∠C are angles
• A, B, and C are vertices

Therefore, every triangle has exactly three angles.

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🔷 Similar Figures

Two figures are called similar if they have:

• Same shape
• Equal corresponding angles
• Corresponding sides in the same ratio

Thus, similar figures may have different sizes but the same shape.

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🔷 Similar Triangles

Two triangles are similar if:

• Their corresponding angles are equal
• Their corresponding sides are proportional

If triangle ABC is similar to triangle DEF, then:$$\triangle ABC \sim \triangle DEF$$△ABC∼△DEF

Therefore:$$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$$DEAB​=EFBC​=DFAC​

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🔷 Criteria for Similarity of Triangles

There are three main criteria.

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🔹 1. AAA Similarity Criterion

If all three corresponding angles are equal, then the triangles are similar.

For example:$$\angle A=\angle D$$∠A=∠D $$\angle B=\angle E$$∠B=∠E $$\angle C=\angle F$$∠C=∠F

Hence, both triangles are similar.

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🔹 2. SAS Similarity Criterion

If two sides are proportional and the included angle is equal, then the triangles are similar.

That means:$$\frac{AB}{DE}=\frac{AC}{DF}$$DEAB​=DFAC​

and$$\angle A=\angle D$$∠A=∠D

Therefore:$$\triangle ABC \sim \triangle DEF$$△ABC∼△DEF

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🔹 3. SSS Similarity Criterion

If all corresponding sides are proportional, then the triangles are similar.

That means:$$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$$DEAB​=EFBC​=DFAC​

As a result, both triangles become similar.

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🔷 Basic Proportionality Theorem (BPT)

This theorem is also called the Thales Theorem.

Statement:

If a line is drawn parallel to one side of a triangle, then it divides the other two sides in the same ratio.

In △ABC, if DE || BC, then:$$\frac{AD}{DB}=\frac{AE}{EC}$$DBAD​=ECAE​

Therefore, parallel lines create proportional segments.

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🔷 Converse of BPT

If a line divides two sides of a triangle in the same ratio, then that line is parallel to the third side.

Thus, proportional sides prove parallelism.

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🔷 Areas of Similar Triangles

If two triangles are similar, then:$$\frac{\text{Area of }\triangle 1}{\text{Area of }\triangle 2} = \left( \frac{\text{Corresponding Side}_1} {\text{Corresponding Side}_2} \right)^2$$Area of △2Area of △1​=(Corresponding Side2​Corresponding Side1​​)2

Hence, area ratio equals the square of side ratio.

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🔷 Pythagoras Theorem

In a right-angled triangle:$$(\text{Hypotenuse})^2=(\text{Base})^2+(\text{Perpendicular})^2$$(Hypotenuse)2=(Base)2+(Perpendicular)2

That means:$$c^2=a^2+b^2$$c2=a2+b2

For example, if a = 3 and b = 4:$$c^2=9+16=25$$c2=9+16=25 $$c=5$$c=5

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🔷 Converse of Pythagoras Theorem

If:$$c^2=a^2+b^2$$c2=a2+b2

then the triangle is right-angled.

Therefore, side lengths can help identify the type of triangle.