NCERT Solutions for Class 10 Mathematics Chapter 6

NCERT solutions for Class 10 Mathematics Chapter Triangles – complete answers & explanations

Triangles is an important chapter in Class 10 Mathematics that helps students understand the concept of similarity, proportionality, and relationships between different sides and angles of triangles. This chapter builds a strong foundation for geometry and is essential for solving real-life problems involving shapes and measurements. These NCERT solutions provide clear and reliable answers that help students learn the correct approach to answering questions step by step. Students can easily understand concepts like similar figures, Basic Proportionality Theorem, and similarity criteria through these solutions. Download the worksheet and practice alongside solutions for better clarity. Book a free trial now to get expert guidance. With consistent practice, students can improve accuracy and gain confidence in solving triangle-based problems.

What this NCERT chapter covers?

1. Understanding the concept of similar figures and how shapes can have the same form but different sizes. 
2. Learning the conditions required for similarity such as equal angles and proportional sides. 
3. Exploring different types of triangles including equilateral and their properties. 
4. Applying the Basic Proportionality Theorem (BPT) to solve numerical problems. 
5. Verifying whether lines are parallel using proportional segments in triangles. 
6. Proving important geometric theorems using logical reasoning and step-by-step methods. 
7. Understanding midpoint theorem and its applications in geometry. 
8. Learning different criteria of triangle similarity like AA, SAS, and SSS. 
9. Solving problems based on proportional sides and corresponding angles. 
10. Applying similarity concepts to find unknown lengths, heights, and distances. 

How to use these NCERT solutions?

1. Start by attempting each question from the worksheet on your own before checking the answers. 
2. Compare your answers with the given solutions to identify mistakes and correct them. 
3. Focus on understanding the steps used in proofs and derivations rather than memorizing them. 
4. Practice similar problems repeatedly to strengthen your concept clarity. 
5. Follow the order of questions exactly as given to maintain proper learning flow. 
6. Use these solutions to revise key theorems and formulas before exams. 
7. Parents and teachers can guide students by discussing each step of the solution. 
8. Pay special attention to reasoning-based questions and similarity proofs. 

Important tips & tricks for students

1. Always check if corresponding angles are equal before concluding similarity. 
2. Ensure that sides are proportional when applying similarity conditions. 
3. Do not skip steps while writing proofs in exams. 
4. Use proper mathematical notation for triangles and ratios. 
5. Revise all similarity criteria (AA, SAS, SSS) thoroughly. 
6. Practice diagram-based questions carefully to avoid calculation mistakes. 
7. Write clear and step-by-step solutions to score full marks. 
8. Understand theorem-based questions instead of memorizing them blindly. 

NCERT solutions – complete answer key

EXERCISE 6.1

(i) similar 
(ii) similar 
(iii) equilateral 
(iv) equal, proportional 

1

(i) Similar figures:
- Two circles (same shape, different size)
- Two squares 

(ii) Non-similar figures:
- Circle and square 
- Rectangle and triangle 

2

Check similarity of quadrilaterals (Fig. 6.8)

Condition for similarity:
1. Corresponding angles equal 
2. Corresponding sides proportional 

If both conditions are NOT satisfied 
⇒ Quadrilaterals are NOT similar 

3
(Using Basic Proportionality Theorem)

(i)
Since DE || BC 

AD/DB = AE/EC 

Substitute values → EC = 3 cm 

(ii)
AD/DB = AE/EC 

Solve → AD = 2 cm 

EXERCISE 6.2
1

Check if EF || QR

Condition:
PE/EQ = PF/FR 

(i)
3.9/3 = 3.6/2.4 
1.3 = 1.5 ❌ Not equal → NOT parallel 

(ii)
4/4.5 = 8/9 
Equal → EF || QR ✔ 

(iii)
0.18/1.28 = 0.36/2.56 
Equal → EF || QR ✔ 

2

Prove:
AM/AB = AN/AD 

Given:
LM || CB and LN || CD 

Using BPT twice:

From LM || CB:
AM/AB = AL/AC ...(1)

From LN || CD:
AN/AD = AL/AC ...(2)

From (1) and (2):

AM/AB = AN/AD ✔ proved 

3

Prove:
BF/FE = BE/EC 

Given:
DE || AC and DF || AE 

Using BPT:

From DF || AE:
BF/FE = BD/DA ...(1)

From DE || AC:
BD/DA = BE/EC ...(2)

From (1) and (2):

BF/FE = BE/EC ✔ proved 

4

Prove EF || QR

Given:
DE || OQ and DF || OR 

Using BPT twice:

From DE || OQ:
OD/DQ = OE/ER 

From DF || OR:
OD/DQ = OF/FR 

So:
OE/ER = OF/FR 

⇒ EF || QR (Converse of BPT) ✔ 

5

Prove BC || QR

Given:
AB || PQ and AC || PR 

Using BPT:

From AB || PQ:
OA/OP = OB/OQ 

From AC || PR:
OA/OP = OC/OR 

So:
OB/OQ = OC/OR 

⇒ BC || QR ✔ 

6

Prove midpoint theorem

If D is midpoint of AB 

AD = DB 

Using BPT:
AE/EC = AD/DB = 1 

⇒ AE = EC 

Line bisects third side ✔ 

7

Prove line joining midpoints is parallel

Given:
AD = DB and AE = EC 

⇒ AD/DB = AE/EC 

By converse of BPT:
DE || BC ✔ 

8

Prove line joining midpoints is parallel

Given:
AD = DB and AE = EC 

⇒ AD/DB = AE/EC Prove:
AO/BO = CO/DO 

9

Using similarity:

ΔAOB ~ ΔCOD 

⇒ AO/BO = CO/DO ✔

By converse of BPT:
DE || BC ✔ 

Prove trapezium

Given:
AO/BO = CO/DO 

⇒ ΔAOB ~ ΔCOD 

⇒ ∠A = ∠C 

⇒ AB || CD 

⇒ ABCD is trapezium ✔ 

10
Similar triangles (based on figure)

Use:
- AAA 
- AA 
- SSS 

Write in symbolic form:
ΔABC ~ ΔDEF 

EXERCISE 6.3
1

Find angles

Given similarity:
ΔODC ~ ΔOBA 

Corresponding angles equal 

Use angle sum property:

∠DOC = 55° 
∠DCO = 55° 
∠OAB = 55° 

2

Prove:
OA/OB = OC/OD 

ΔAOB ~ ΔCOD (AA similarity)

⇒ corresponding sides proportional ✔ 

3

Prove:
ΔPQS ~ ΔTQR 

Given:
QR/QS = QT/PR and ∠1 = ∠2 

⇒ SAS similarity 

✔ Triangles similar 

4

Prove:
ΔRPQ ~ ΔRTS 

Given:
∠P = ∠RTS 

Using angle equality + common angle 

⇒ AA similarity ✔ 

5

Prove:
ΔADE ~ ΔABC 

Given:
ΔABE ≅ ΔACD 

⇒ AB = AC and ∠ equal 

⇒ corresponding sides proportional 

⇒ similarity ✔ 

6

Multiple similarity proofs

(i) ΔAEP ~ ΔCDP (AA) 
(ii) ΔABD ~ ΔCBE (AA) 
(iii) ΔAEP ~ ΔADB (AA) 
(iv) ΔPDC ~ ΔBEC (AA) 

7

Prove:
ΔABE ~ ΔCFB 

Using angle equality + parallel lines 

⇒ AA similarity ✔ 

8

Prove:

(i) ΔABC ~ ΔAMP 

Both right triangles 
Common angle 

⇒ AA similarity ✔ 

(ii)
CA/PA = BC/MP 

From similarity ✔ 

9

Prove:

(i)
CD/AC = GH/FG 

Using similarity ✔ 

(ii) ΔDCB ~ ΔHGE 
(iii) ΔDCA ~ ΔHGF 

10

Prove:
ΔABD ~ ΔECF 

Using:
Right angles + angle equality 

⇒ AA similarity ✔ 

11

Prove:
ΔABC ~ ΔPQR 

Given proportional sides 

⇒ SSS similarity ✔ 

12

Prove:
CA² = CB × CD 

Using similarity:

ΔADC ~ ΔBAC 

13

⇒ CA/CB = CD/CA 

⇒ CA² = CB × CD ✔ 

Prove:
ΔABC ~ ΔPQR 

Using:
Median proportional 

⇒ SAS similarity ✔ 

14

Height of tower

Using similarity:

6/4 = h/28 

h = 42 m ✔ 

15

Prove:
AB/PQ = AD/PM 

Using similarity of triangles and medians 

⇒ proportional sides ✔ 
 

Why NCERT solutions help students?

NCERT solutions help students understand the correct method of solving problems, improve conceptual clarity, and prepare effectively for exams. They ensure that answers follow the expected format, making it easier to score better marks and build confidence in Mathematics.

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