Class 9 Maths – Triangles

7.1 Introduction

A triangle is a closed figure formed by joining three non-collinear points with line segments.
Vertices: The three points (A, B, C)
Sides: Line segments joining the vertices (AB, BC, CA)
Angles: ∠A, ∠B, ∠C

Importance: Triangles are fundamental in geometry, construction, and trigonometry.

Congruent triangles: Triangles that are exactly the same in shape and size.
Symbols: ΔABC ≅ ΔPQR

Two triangles are congruent if:

SSS (Side-Side-Side):
- If all three sides of one triangle are equal to the corresponding sides of another, triangles are congruent.
SAS (Side-Angle-Side):
- Two sides and the included angle are equal.
ASA (Angle-Side-Angle):
- Two angles and the included side are equal.
AAS (Angle-Angle-Side):
- Two angles and a non-included side are equal.
RHS (Right angle-Hypotenuse-Side):(for right-angled triangles)
- Right angle, hypotenuse, and one side equal → triangles congruent

Sum of angles in a triangle: 180°
Exterior angle property: Exterior angle = Sum of two opposite interior angles
Length property: Sum of any two sides > third side

Using midpoints, altitudes, medians for proving congruence in complex problems
Often applied in coordinate geometry and construction problems

Question	Short Answer
Sum of angles in a triangle?	180°
Definition of congruent triangles?	Same shape & size
SSS criterion?	Three sides equal
SAS criterion?	Two sides + included angle equal
ASA criterion?	Two angles + included side equal
AAS criterion?	Two angles + non-included side equal
RHS criterion?	Right angle, hypotenuse & side equal
Exterior angle property?	Exterior = sum of opposite interior angles
Triangle sides property?	Sum of any two sides > third side
Symbol for congruence?	ΔABC ≅ ΔPQR