📐 Triangles - Class 9

Complete notes with congruence criteria, theorems, and formula sheet

1. Introduction to Triangles

A triangle is a closed figure formed by three line segments. It's one of the most fundamental shapes in geometry with numerous properties and applications.

📖 What is a Triangle?

A triangle is a polygon with three sides, three vertices, and three angles.

The sum of all three angles is always 180°.

Notation: Triangle ABC is written as △ABC.

2. Types of Triangles

2.1 Based on Sides

Type	Properties	Example
Scalene	All three sides different	Sides: 3, 4, 5 cm
Isosceles	Two sides equal	Sides: 5, 5, 7 cm
Equilateral	All three sides equal	Sides: 6, 6, 6 cm

2.2 Based on Angles

Acute Triangle: All three angles less than 90°.
Right Triangle: One angle equals 90°.
Obtuse Triangle: One angle greater than 90°.

3. Congruence of Triangles

📖 Congruent Triangles

Two triangles are congruent if they have the same shape and size.

All corresponding sides and angles are equal.

Symbol: ≅ (△ABC ≅ △PQR means triangle ABC is congruent to triangle PQR)

3.1 Criteria for Congruence

⚠️ SAS (Side-Angle-Side)

Two triangles are congruent if two sides and the included angle of one triangle are equal to corresponding two sides and included angle of the other.

⚠️ ASA (Angle-Side-Angle)

Two triangles are congruent if two angles and the included side of one triangle are equal to corresponding two angles and included side of the other.

⚠️ AAS (Angle-Angle-Side)

Two triangles are congruent if any two angles and a non-included side of one triangle are equal to corresponding angles and side of the other.

⚠️ SSS (Side-Side-Side)

Two triangles are congruent if all three sides of one triangle are equal to corresponding three sides of the other.

⚠️ RHS (Right angle-Hypotenuse-Side)

Two right triangles are congruent if the hypotenuse and one side of one triangle are equal to hypotenuse and corresponding side of the other.

🔑 Important Notes on Congruence

SSA (Side-Side-Angle) is NOT a valid congruence criterion
AAA (Angle-Angle-Angle) proves similarity, not congruence
Order matters when writing congruence: AB = PQ, BC = QR, CA = RP
Corresponding parts of congruent triangles are equal (CPCT)

📝 Example: Congruence

Q: In △ABC and △PQR, AB = PQ, BC = QR, and ∠B = ∠Q. Are they congruent?

Solution:

Given: AB = PQ, BC = QR, ∠B = ∠Q

This matches SAS criterion (two sides and included angle)

Yes, △ABC ≅ △PQR by SAS

4. Properties of Triangles

4.1 Angle Sum Property

Sum of three angles = 180°
If ∠A, ∠B, ∠C are angles: ∠A + ∠B + ∠C = 180°

4.2 Exterior Angle Property

Exterior angle = Sum of two opposite interior angles
Greater than either non-adjacent interior angle

4.3 Inequalities in Triangles

Triangle Inequality: Sum of any two sides > Third side
If a > b > c, then ∠A > ∠B > ∠C (greater side has greater opposite angle)
Difference of any two sides < Third side

📚 Quick Formula Sheet - Triangles

Congruence Criteria

SAS: Side-Angle-Side

ASA: Angle-Side-Angle

AAS: Angle-Angle-Side

SSS: Side-Side-Side

RHS: Right-Hypotenuse-Side

Triangle Properties

Angle sum = 180°

Exterior angle = sum of opposite interiors

Sum of 2 sides > 3rd side

Fundamental theorems

Types by Sides

Scalene: All different

Isosceles: 2 equal

Equilateral: All equal

Classification

Types by Angles

Acute: All < 90°

Right: One = 90°

Obtuse: One > 90°

Based on angles

💡 Study Tips

• Memorize all five congruence criteria - they're the foundation!

• Draw diagrams for every problem - visualization is key.

• Mark equal sides/angles clearly on diagrams.

• CPCT (Corresponding Parts of Congruent Triangles) is very useful in proofs.

• Remember: SSA is NOT valid, but RHS is (special case for right triangles).

🔑 Common Mistakes

Don't confuse congruence with similarity
Remember included angle means angle between the two sides
Order matters in congruence statements
SSA doesn't work - avoid using it!