THE GOAL | Class 10 Maths | Chapter 6

01. Similar Figures

Two figures are called similar if they have the same shape, but their sizes may be different. For similar figures:

• Corresponding angles are equal.
• Corresponding sides are proportional.

💡 Easy Hindi: Similarity ka simple meaning hai: Shape same, size different ho sakta hai.

✏ Example

If two triangles have angles:
40°, 60°, 80°

and another triangle also has:
40°, 60°, 80°

then the triangles have the same shape and are similar.

02. Similar Triangles

Two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional.

△ABC ∼ △DEF

Therefore:

AB DE

=

BC EF

=

CA FD

💡 Correspondence:

A ↔ D
B ↔ E
C ↔ F

Similarity statement ka order hi corresponding vertices batata hai.

03. Basic Proportionality Theorem — BPT ⭐⭐⭐

Statement: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

AD DB

=

AE EC

💡 Hindi: Triangle ke andar agar ek line kisi side ke parallel draw ki jaaye, toh baaki dono sides same ratio mein divide hoti hain.

Parallel → Proportion

04. BPT — Solved Example

✏ Example

In △ABC, DE ∥ BC. Given:
AD = 3 cm, DB = 2 cm, AE = 6 cm. Find EC.

By BPT:

AD DB

=

AE EC

Therefore:

3 2

=

6 x

3x = 12
x = 4 cm

✓ EC = 4 cm

05. BPT — Proof ⭐⭐⭐

Given:
In △ABC, D lies on AB and E lies on AC such that
DE ∥ BC.

To Prove:

AD DB

=

AE EC

Proof:

Join BE and CD.

Triangles △ADE and △BDE have the same altitude from E to AB. Therefore,

ar(△ADE) ar(△BDE)

=

AD DB

Similarly, △ADE and △CDE have the same altitude from D to AC. Therefore,

ar(△ADE) ar(△CDE)

=

AE EC

Since DE ∥ BC, triangles △BDE and △CDE are on the same base DE and between the same parallels DE and BC. Therefore,

ar(△BDE) = ar(△CDE)

Hence,

AD DB

=

AE EC

Hence proved.

💡 Proof ka idea: Same altitude wale triangles ke areas ka ratio unki corresponding bases ke ratio ke equal hota hai. Isi area relation se BPT ka result milta hai.

06. Converse of BPT ⭐⭐⭐

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

💡 BPT ka direction: Parallel → Ratio

Converse ka direction: Ratio → Parallel

✏ Example

AD = 4 cm
DB = 6 cm
AE = 6 cm
EC = 9 cm

AD DB

=

4 6

=

2 3

and

AE EC

=

6 9

=

2 3

Therefore:

DE ∥ BC by converse of BPT.

07. AA Similarity Criterion ⭐⭐⭐

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

∠A = ∠D
∠B = ∠E

Therefore:

△ABC ∼ △DEF

by AA similarity criterion.

💡 Do corresponding angles equal mil jaayein, toh third angles automatically equal honge because the sum of angles of a triangle is 180°.

✏ Example

In △ABC and △PQR:
∠A = ∠P = 50°
∠B = ∠Q = 60°

Then:
∠C = ∠R = 70°

Hence:

△ABC ∼ △PQR

by AA similarity.

08. SSS Similarity Criterion ⭐⭐⭐

If the corresponding sides of two triangles are proportional, then the triangles are similar.

3 6

=

4 8

=

5 10

=

1 2

Therefore:

△ABC ∼ △PQR

💡 SSS mein three corresponding sides proportional honi chahiye. Angles find karna necessary nahi hai.

09. SSS — Solved Example

✏ Example

Check whether triangles with sides
3 cm, 4 cm, 5 cm
and
6 cm, 8 cm, 10 cm
are similar.

3/6 = 1/2
4/8 = 1/2
5/10 = 1/2

All corresponding sides are proportional.

Therefore, the triangles are similar by SSS.

10. SAS Similarity Criterion ⭐⭐⭐

If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the triangles are similar.

∠A = ∠D

AB DE

=

AC DF

Therefore:

△ABC ∼ △DEF

by SAS similarity.

💡 Important: SAS mein angle wahi hona chahiye jo given dono sides ke beech mein ho. Isko included angle kehte hain.

11. SAS — Solved Example

✏ Example

Given:
AB = 4 cm, AC = 6 cm
DE = 8 cm, DF = 12 cm
and ∠A = ∠D. Check similarity.

AB DE

= 4/8 = 1/2

AC DF

= 6/12 = 1/2

∠A = ∠D

Therefore △ABC ∼ △DEF by SAS.

12. Similar Triangles — Find Missing Side ⭐⭐⭐

✏ Example

Given:
△ABC ∼ △PQR
AB = 6 cm
PQ = 9 cm
BC = 8 cm
Find QR.

Since:

△ABC ∼ △PQR

Corresponding sides:

AB PQ

=

BC QR

Therefore:

6 9

=

8 x

6x = 72
x = 12 cm

✓ QR = 12 cm

13. BPT Application — Unknown Length

✏ Example

In △ABC, DE ∥ BC. AD = 5 cm
DB = 3 cm
AE = 10 cm
Find EC.

By BPT:

AD DB

=

AE EC

5 3

=

10 x

5x = 30
x = 6 cm

✓ EC = 6 cm

14. Corresponding Sides — Very Important

If:

△ABC ∼ △DEF

Then:
A ↔ D
B ↔ E
C ↔ F

Therefore:

AB DE

=

BC EF

=

CA FD

⚠ Common mistake: Similarity statement ka order ignore karke sides ko random compare mat karo.

15. Which Similarity Criterion?

Given in Question	Criterion	What to Check
Two corresponding angles equal	AA	2 angles
Three corresponding sides proportional	SSS	3 side ratios
Two sides proportional + included angle equal	SAS	2 sides + included angle

💡 Question mein jo information actually given hai, usi ke according criterion select karo.

16. BPT vs Converse — Don't Confuse

BPT

DE ∥ BC

↓

AD DB

=

AE EC

Converse

AD DB

=

AE EC

↓

DE ∥ BC

17. How to Write a Similarity Proof

Step 1: Identify the two triangles.

Step 2: Write the equal angles or proportional sides.

Step 3: Select AA / SSS / SAS.

Step 4: Write the similarity statement in correct order.

Step 5: Use corresponding sides if a length has to be found.

💡 Exam approach: Pehle triangles identify karo → correspondence set karo → criterion lagao → similarity prove karo → phir required ratio use karo.

18. Common Exam Mistakes ⚠️

1. Similarity statement ka order galat likhna.

2. Corresponding sides ko opposite order mein compare karna.

3. BPT mein DE ∥ BC condition miss kar dena.

4. Converse mein ratios equal establish kiye bina parallel conclude kar dena.

5. SAS mein non-included angle use karna.

6. SSS mein sides ka correspondence incorrectly match karna.

7. Calculation ke baad unit na likhna.

19. One-Page Revision ⭐⭐⭐

Similarity
Same shape, size may differ.

BPT

DE ∥ BC ⇒

AD DB

=

AE EC

Converse of BPT

AD DB

=

AE EC

⇒ DE ∥ BC

AA
Two corresponding angles equal ⇒ Similar triangles

SSS
Three corresponding sides proportional ⇒ Similar triangles

SAS
Two corresponding sides proportional + included angle equal ⇒ Similar triangles

20. Final Exam Check 🎯

✓ Can you identify similar triangles?

✓ Can you write correct correspondence?

✓ Can you apply BPT?

✓ Can you apply converse of BPT?

✓ Can you prove similarity by AA?

✓ Can you prove similarity by SSS?

✓ Can you prove similarity by SAS?

✓ Can you find an unknown side using proportional sides?

✓ Can you write a proper proof step-by-step?

💡 Chapter ka core idea:

Triangle → Similarity → Proportion → Unknown Length

Aur BPT ke liye:
Parallel line → Same ratio

Converse ke liye:
Same ratio → Parallel line

△ TRIANGLES

Class 10 Mathematics • Chapter 6