Triangles Class 9 ICSE — Chapter 9

Written by Greya LakshmiPublished on 12 October 2026

9.1 INTRODUCTION

The Triangles Class 9 ICSE chapter opens with the definition you will use in every proof that follows. A triangle is a plane figure bounded by three straight line segments. It has three vertices and three sides. The chapter then links sides to angles, introduces medians and altitudes, and finishes with the five conditions that prove two triangles congruent.

Download Triangles Class 9 ICSE Mathematics Chapter 9 PDF

Download the Triangles Class 9 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 13 pages, 5 sections, 29 questions.

What is inside Chapter 9

13 pages · 5 sections · 8 worked examples · 29 exercise questions

Section	Title
9.1	INTRODUCTION
9.2	RELATION BETWEEN SIDES AND ANGLES OF TRIANGLES
9.3	SOME IMPORTANT TERMS
9.4	CONGRUENT TRIANGLES
9.5	CONDITIONS FOR CONGRUENCY OF TRIANGLES

EXERCISE 9(A): 15 questions
EXERCISE 9(B): 14 questions

For a triangle named ABC, written $\Delta ABC$ , the three vertices are A, B and C. The three sides are AB, BC and CA. Each side is named by the two vertices it joins. That naming matters throughout the chapter, because later sections speak of the angle opposite side AB and the side opposite angle C. Keep those pairs clear from the start.

The definition covers every triangle in the chapter, and the congruency proofs in the exercises rely on it. For the syllabus that frames this chapter, check the official CISCE website.

9.2 RELATION BETWEEN SIDES AND ANGLES OF TRIANGLES

This section states four rules that connect the lengths of sides to the measures of the angles opposite them. Rule one: if all three sides have different lengths, all three angles have different measures, and the greater side has the greater angle opposite it. Rule two is the converse: if the angles are all different, the sides are all different, and the greater angle has the greater side opposite it.

A longer side forces the two ends of the opposite angle further apart, so that angle opens wider.

Rule three covers equal parts. If two sides are equal, the angles opposite them are equal; if two angles are equal, the sides opposite them are equal. Rule four extends this to all three sides: equal sides give equal angles, and equal angles give equal sides.

Worked pattern. In a triangle with sides 6 cm, 8 cm and 10 cm, let $AB = 6\ \text{cm}$ , $BC = 8\ \text{cm}$ and $CA = 10\ \text{cm}$ . Side CA is the greatest, so the angle opposite CA, which is $\angle B$ , is the greatest. Side AB is the smallest, so $\angle C$ is the smallest.

The order of angles is $\angle B \gt \angle A \gt \angle C$ .

9.3 SOME IMPORTANT TERMS

A median of a triangle, corresponding to any side, is the line joining the midpoint of that side to the opposite vertex. Each triangle has three medians, and the three medians always meet at a single point called the centroid. The centroid divides every median in the ratio 2:1, with the longer part towards the vertex. For the median AD, with D the midpoint of BC and G the centroid, $AG : GD = 2 : 1$ .

An altitude, corresponding to any side, is the perpendicular drawn from the opposite vertex to that side. Every triangle has three altitudes, and all three pass through one point called the orthocentre. The altitude measures the shortest distance from a vertex to its opposite side.

Two angle facts carry most of this chapter. First, the sum of the angles of a triangle equals two right angles, $180^\circ$ . Second, if one side is produced, the exterior angle so formed equals the sum of the two interior opposite angles. At vertex A, the exterior angle is $\angle B + \angle C$ . The exterior angle and its adjacent interior angle form a straight line, so the equality follows from the $180^\circ$ sum.

The section closes with six corollaries drawn from these facts:

An exterior angle is greater than each of the interior opposite angles.
A triangle cannot have more than one right angle.
A triangle cannot have more than one obtuse angle.
In a right-angled triangle, the sum of the other two angles is $90^\circ$ .
In every triangle, at least two angles are acute.
If two angles of one triangle equal two angles of another, each to each, the third angles are also equal.

9.4 CONGRUENT TRIANGLES

Two triangles are congruent if, on placing one over the other, they coincide exactly. Congruent triangles have the same shape and the same size. Same shape means each angle of one triangle equals the corresponding angle of the other; same size means each side of one equals the corresponding side of the other. The symbol $\cong$ is read as “is congruent to”.

Corresponding parts are fixed by position. The corresponding sides lie opposite the equal angles, and the corresponding angles lie opposite the equal sides. In $\Delta ABC$ and $\Delta DEF$ , if $\angle A = \angle D$ , then the sides opposite them, BC and EF, are corresponding sides.

Once congruence is proved, any remaining pair of matching parts can be declared equal. This is the CPCTC rule: corresponding parts of congruent triangles are also congruent. It is the step that turns a proved congruence into the side or angle equality a question asks for.

9.5 CONDITIONS FOR CONGRUENCY OF TRIANGLES

Five conditions decide congruence without checking all six pairs of parts. SSS needs three equal sides. SAS needs two sides and the included angle — if the equal angle is not between them, the triangles need not be congruent. ASA needs two angles and the included side. AAS needs two angles and a corresponding side. RHS is for right triangles only: hypotenuse and one side equal.

Condition	Parts that must match	Type of triangle
SSS	all three sides	any triangle
SAS	two sides and the included angle	any triangle
ASA	two angles and the included side	any triangle
AAS	two angles and a side opposite one equal angle	any triangle
RHS	right angle, hypotenuse, one side	right triangles only

The AAS rule carries a precaution. The equal sides must be corresponding sides, meaning each side must be opposite to one of the equal angles. In triangles PQR and STU, suppose $\angle P = \angle S$ and $\angle Q = \angle T$ . Side QR, opposite angle P, must equal side TU, opposite angle S.

If the given equal sides are PR and TU instead, they are not opposite to equal angles, so the triangles are not congruent by AAS.

AAS is another form of ASA. Suppose the equal side is $PR = SU$ , with $\angle P = \angle S$ and $\angle Q = \angle T$ . Since the angles of every triangle sum to $180^\circ$ , the third angles are forced to be equal: $\angle R = \angle U$ .

Now side PR lies between angle P and angle R, while side SU lies between angle S and angle U. The pair therefore satisfies ASA with $\angle P = \angle S$ , $\angle R = \angle U$ , and $PR = SU$ .

The order of vertices in a congruence statement matters. If A corresponds to D, B to E and C to F, write $\Delta ABC \cong \Delta DEF$ . Writing $\Delta ABC \cong \Delta DFE$ or $\Delta BAC \cong \Delta DEF$ is wrong, because those orders claim the wrong vertices correspond.

Worked pattern using SAS. In $\Delta PQR$ and $\Delta XYZ$ , let $PQ = XY = 5\ \text{cm}$ , $QR = YZ = 7\ \text{cm}$ , and $\angle PQR = \angle XYZ = 60^\circ$ .

Step 1: Confirm the equal angle lies between the equal sides in both triangles. It does in each.

$\Delta PQR \cong \Delta XYZ \quad [katex display="true"]\text{By S.A.S.}$ [/katex]

Final answer: By CPCTC, $PR = XZ$ , $\angle QPR = \angle YXZ$ , and $\angle PRQ = \angle XZY$ .

The same congruence rules return in the isosceles triangles chapter, where equal sides and equal angles meet again.

What to practise from Triangles Class 9 ICSE

Both exercise blocks in this chapter drill the congruency conditions and the CPCTC step that finishes a proof. Name the rule you use at every step, and state which vertices correspond.

EXERCISE 9(A) — 15 questions on congruence of triangles, including circle-based and parallelogram proofs.
EXERCISE 9(B) — 14 questions on further congruence proofs with squares, parallelograms, medians and equilateral triangles.

More Class 9 Maths material, chapter by chapter, lives on the Class 9 Maths chapter hub.

Triangles Class 9 ICSE frequently asked questions

What is the difference between ASA and AAS in triangles class 9 ICSE?

ASA needs two angles and the side between them. AAS needs two angles and a side opposite one of those angles. They are linked: since the angles of a triangle sum to $180^\circ$ , the third angles match too, so every AAS case can be rewritten as ASA.

How do I write a congruence statement for triangles in Class 9 ICSE?

Write the vertices in matching order. If A corresponds to D, B to E and C to F, write $\Delta ABC \cong \Delta DEF$ . Wrong orders such as $\Delta ABC \cong \Delta DFE$ claim false correspondences.

Which congruency rule should I use in Class 9 ICSE triangle proofs?

Match the rule to the given data. Three sides give SSS. Two sides with the included angle give SAS. Two angles with the included side give ASA. Two angles with a corresponding side give AAS. A right triangle with hypotenuse and one side gives RHS.

What does CPCTC mean in the Class 9 ICSE Triangles chapter?

It stands for Corresponding Parts of Congruent Triangles are Congruent. After you prove two triangles congruent, CPCTC lets you state that any remaining pair of matching sides or angles is equal.

How many exercises are there in the Class 9 ICSE Triangles chapter?

Two. Exercise 9(A) has 15 questions, and Exercise 9(B) has 14 questions.

Reference: ICSE Class 9 Mathematics textbook material, chapter 9: Triangles — CISCE-aligned.

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