Similarity Class 10 ICSE — Bansal Chapter 15

Written by Greya LakshmiPublished on 28 September 2026

Similarity class 10 ICSE is the study of figures that share the same shape but may differ in size. This chapter from the Bansal Mathematics book covers similar triangles, the SAS, AA and SSS criteria, the basic proportionality theorem, and area ratios. It ends with size transformations and their use in maps and models. Each section below follows the book’s own numbering, with definitions and formulas explained in plain words.

Download the Similarity Class 10 ICSE Mathematics chapter PDF from the Bansal textbook to revise offline or print for practice. It is the full chapter as printed in the book — 27 pages, 8 sections, 52 questions.

What is inside Chapter 15

27 pages · 8 sections · 15 worked examples · 52 exercise questions

Section	Title
15.1	Introduction
15.2	Similar Triangles
15.3	Corresponding sides and corresponding angles
15.4	Condition of Similar Triangle : (SAS, AA or AAA and SSS)
15.5	Basic Proportionality Theorem With Applications
15.6	Relation between the areas of two triangles
15.7	Similarity as a Size Transformation
15.8	Applications to Maps and Models

EXERCISE 15(B): 10 questions
EXERCISE 15(D): 8 questions
EXERCISE 15(E): 34 questions

15.1 Introduction

The chapter opens with two definitions. Similar figures have the same shape but may differ in size. Congruent figures have the same shape and the same size.

Same shape means each angle of one figure equals the corresponding angle of the other. Same size means each side of one figure equals the corresponding side of the other.

The relationship between the two ideas matters. Congruent figures are always similar, because equal size forces proportional sides. But similar figures are not necessarily congruent, because their sizes may differ.

Property	Similar figures	Congruent figures
Shape	Same	Same
Size	May differ	Must match
Relationship	Not always congruent	Always similar

Think of a photograph printed in two sizes. Both prints have identical shapes, so they are similar. They are not congruent unless the prints are the same size. This distinction runs through the entire Class 10 ICSE syllabus.

15.2 Similar Triangles

Two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional. Both conditions must hold together.

For triangles ABC and PQR, similarity means three angle equalities and one side ratio chain:

\angle A = \angle P, \angle B = \angle Q, \angle C = \angle R

and

\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}

The symbol $\sim$ is read as “is similar to”. So we write $\Delta ABC \sim \Delta PQR$ .

Here is a fresh numerical check. Take a triangle with sides 4, 6 and 8 units. Compare it with a triangle with sides 6, 9 and 12 units.

\frac{4}{6} = \frac{6}{9} = \frac{8}{12} = \frac{2}{3}

All three ratios are equal, so the triangles are similar. The common scale factor is $\frac{2}{3}$ .

15.3 Corresponding sides and corresponding angles

Once two triangles are similar, you must identify which sides and angles correspond. Sides opposite to equal angles are the corresponding sides. If $\angle A = \angle P$ , then the sides opposite them, BC and QR, correspond.

The second rule reverses the direction. Angles opposite to proportional sides are equal. If $\frac{AB}{EF} = \frac{BC}{DF}$ , the angles opposite those sides match up too.

There is a sharp contrast with congruent triangles. In congruent triangles, corresponding sides are equal in length. In similar triangles, corresponding sides are only in proportion.

One more result is worth keeping. Triangles that are similar to the same triangle are similar to each other. In this ICSE chapter, that transitive property appears in many proofs.

Here is a matching exercise with fresh labels. Let $\Delta XYZ \sim \Delta LMN$ with $\angle X = \angle L$ , $\angle Y = \angle M$ and $\angle Z = \angle N$ . Then side YZ corresponds to MN, side XZ corresponds to LN, and side XY corresponds to LM.

15.4 Condition of Similar Triangle : (SAS, AA or AAA and SSS)

Three postulates prove two triangles similar without checking everything. Each one is a shortcut.

SAS: one angle of one triangle equals one angle of the other, and the sides including those angles are in proportion.
AA or AAA: at least two pairs of corresponding angles are equal. The third pair follows because angles of a triangle sum to $180^\circ$ .
SSS: all three pairs of corresponding sides are in proportion.

The vertex order rule is easy to get wrong. In $\Delta ABC \sim \Delta DEF$ , vertex A corresponds to D, B to E and C to F. The order of vertices must match the order of equal angles.

For a right-angled triangle with altitude AD drawn to the hypotenuse BC, the chapter proves a powerful result:

\Delta DBA \sim \Delta DAC \sim \Delta ABC

This yields three squared-side relations:

$AB^2 = BD \times BC$ , $AC^2 = DC \times BC$ , $AD^2 = BD \times DC$

Fresh numbers: let BD = 4 cm and DC = 9 cm. Then $AD^2 = 4 \times 9 = 36$ , so AD = 6 cm. The altitude is the geometric mean of the two hypotenuse segments. Solved examples in this section carry board years 2000, 2005 and 2012.

15.5 Basic Proportionality Theorem With Applications

Theorem 1 states: a line drawn parallel to one side of a triangle divides the other two sides proportionally.

In $\Delta ABC$ , if DE is parallel to BC, with D on AB and E on AC, then:

\frac{AD}{DB} = \frac{AE}{EC}

The proof uses similarity. Since DE is parallel to BC, the angles at D and E match corresponding angles of $\Delta ABC$ . So $\Delta ADE \sim \Delta ABC$ , and the side ratios follow.

The converse also holds. If a line divides two sides of a triangle proportionally, that line is parallel to the third side.

Fresh example: let D divide AB in the ratio 2:3. Then $\frac{AE}{EC} = \frac{2}{3}$ . If BC = 7.5 cm, similarity gives $\frac{DE}{BC} = \frac{AD}{AB} = \frac{2}{5}$ , so DE = 3 cm.

This theorem reappears across the Class 10 ICSE Maths curriculum. Exercise 15(B) has 10 questions that apply it and its converse.

15.6 Relation between the areas of two triangles

Theorem 2 states: the areas of two similar triangles are proportional to the squares of their corresponding sides.

If $\Delta ABC \sim \Delta DEF$ , then:

\frac{\text{Area of } \Delta ABC}{\text{Area of } \Delta DEF} = \frac{AB^2}{DE^2} = \frac{BC^2}{EF^2} = \frac{AC^2}{DF^2}

The proof draws altitudes from corresponding vertices and shows the altitude ratio equals the side ratio.

Two reminder facts support later problems. A median divides a triangle into two equal-area triangles. Triangles sharing a vertex with bases on one straight line have areas in the ratio of their base lengths.

Fresh area example: two similar triangles have corresponding sides in ratio 1:3. Their areas are in ratio $1^2 : 3^2 = 1:9$ . A side of 5 cm against 15 cm gives areas in ratio 25:225, which simplifies to 1:9.

Board questions from 2004 and 2007 appear in the solved examples of this section.

15.7 Similarity as a Size Transformation

Draw triangle ABC and mark a point P outside it. Join PA, PB and PC. Produce each to double the distance, so PA’ = 2PA, PB’ = 2PB and PC’ = 2PC. Joining A’, B’ and C’ gives a triangle twice the size of ABC.

This is an enlargement by scale factor 2 about centre P. If instead you take PA’ = ½PA, PB’ = ½PB and PC’ = ½PC, you get a reduction by scale factor ½.

Both operations are size transformations. The image is similar to the object. The transformation preserves the kind of figure: the image of a line is a line, of a triangle a triangle, of a quadrilateral a quadrilateral.

The construction works with any polygon. The centre P stays fixed while every point moves along a ray from P.

The scale factor k decides the effect. If k is greater than 1, you have an enlargement. If k is less than 1, you have a reduction. If k equals 1, the figure is unchanged, an identity transformation.

15.8 Applications to Maps and Models

A map’s scale factor k is the ratio between a distance on the map and the actual distance on the ground. A scale of 1:20,000 means 1 cm on the map equals 20,000 cm, or 0.2 km, on the ground.

The same idea applies to models. Height, length and width of the model all equal k times the corresponding dimension of the object.

Three rules govern every problem:

sides scale by $k$
areas scale by $k^2$
volumes scale by $k^3$

Fresh map example: a map of scale 1:50,000 shows a plot side of 3.5 cm. The actual length is $3.5 \times 50{,}000 = 175{,}000$ cm, which is 1.75 km.

Fresh model example: a ship model at scale 1:200 has length 4 m. The ship is $200 \times 4 = 800$ m long. If the model’s volume is 200 litres, the ship’s volume is $200^3 = 8{,}000{,}000$ times that.

These scale rules follow the pattern set in the official CISCE syllabus for map and model problems. Exercise 15(D) has 8 questions on this topic. Exercise 15(E) has 34 questions covering the whole chapter.

What to practise from Similarity Class 10 ICSE

Three exercise blocks make up the chapter’s practice. Work through them in order after finishing each section.

EXERCISE 15(B): 10 questions on the basic proportionality theorem and its converse.
EXERCISE 15(D): 8 questions on maps, models and scale factors.
EXERCISE 15(E): 34 questions revising the whole chapter.

Solved examples carry board years that map to specific sections. Section 15.4 holds questions from 2000, 2005 and 2012. Section 15.6 holds questions from 2004 and 2007. Exercise 15(E) holds questions from 2011, 2013 and 2015. The chapter works as a question bank for the Class 10 ICSE board exam.

Similarity Class 10 ICSE Frequently Asked Questions

What is similarity in Class 10 ICSE and how does it differ from congruency?

Similar figures have the same shape but may differ in size. Congruent figures have the same shape and the same size. Every congruent figure is similar, but not every similar figure is congruent.

Which sides correspond in similar triangles and how do I write the similarity statement?

Sides opposite equal angles correspond, and angles opposite proportional sides are equal. Write the vertices in matching order, so $\Delta ABC \sim \Delta DEF$ means A matches D, B matches E and C matches F.

When do I use the SAS, AA and SSS criteria to prove two triangles similar?

Use SAS when one angle matches and the two sides around it are proportional. Use AA when two angle pairs match. Use SSS when all three side pairs are proportional.

What does the basic proportionality theorem state and how is the converse used?

A line parallel to one side of a triangle divides the other two sides proportionally. The converse says that if a line divides two sides proportionally, it is parallel to the third side. Both forms appear in Exercise 15(B).

How does the scale factor change side lengths, areas and volumes in map and model problems?

Sides scale by $k$ , areas by $k^2$ , and volumes by $k^3$ . If $k \gt 1$ you have an enlargement, if $k \lt 1$ a reduction, and if $k = 1$ the figure is unchanged.

Reference: ICSE Class 10 Mathematics textbook material, chapter 15: Similarity — CISCE-aligned.

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