NCERT Solutions for Class 10 Mathematics Chapter Triangles

This worksheet provides NCERT Solutions for Class 10 Mathematics Chapter Triangles. This chapter focuses on understanding similarity of figures, properties of triangles, and important theorems such as the Basic Proportionality Theorem. It helps students learn how to compare shapes, prove relationships, and solve geometry problems using logical reasoning. This worksheet provides complete and accurate NCERT Solutions that strictly follow the NCERT pattern, helping students build a strong foundation in geometry.

Chapter summary: stories, poems & themes

This chapter is concept-based and does not include stories or poems. It focuses on geometric figures, diagrams, and activity-based problem solving. Students learn about similarity of figures, proportional relationships, and triangle properties through diagrams and proofs. The main theme of the chapter is understanding how shapes relate to each other using mathematical logic and visual reasoning.

What this NCERT chapter covers?

• Understanding similar and non-similar figures 
• Conditions for similarity of triangles and quadrilaterals 
• Basic Proportionality Theorem (BPT) and its applications 
• Converse of BPT and its use in proofs 
• Midpoint theorem and parallel line properties 
• Triangle similarity criteria such as AA, SAS, and SSS 
• Solving geometry problems using ratios and proportions 5

How to use these NCERT solutions?

Students should first try solving each question on their own before checking the answers. These solutions follow the exact NCERT order and structure, making it easy to match questions with answers. Parents and teachers can use this worksheet to guide students step by step and ensure clarity of concepts. Regular practice using these solutions helps in revision and strengthens problem-solving skills.

Student tips & learning tricks

• Always check if corresponding angles are equal before proving similarity 
• Carefully apply the correct similarity rule (AA, SAS, or SSS) 
• Do not skip steps while writing proofs 
• Use proper ratios when applying the Basic Proportionality Theorem 
• Practice diagram-based questions to improve understanding 
• Recheck calculations when solving proportional equations 

Why NCERT solutions are important?

NCERT Solutions help students understand concepts in a clear and structured way. They follow the CBSE curriculum closely and ensure accurate learning. These solutions improve confidence, help in exam preparation, and strengthen the foundation in Mathematics. Regular practice with NCERT-based questions prepares students for school assessments and competitive exams.

Complete answer key – NCERT solutions

Exercise 6.1

1. 
(i) similar 
(ii) similar 
(iii) equilateral 
(iv) equal, proportional 

2. 
(i) Similar figures: 
- Two circles (same shape, different size) 
- Two squares 

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

3. 
Condition for similarity: 
1. Corresponding angles equal 
2. Corresponding sides proportional 

If both conditions are NOT satisfied 
⇒ Quadrilaterals are NOT similar 

Exercise 6.2

1. 
(i) 
Since DE || BC 
AD/DB = AE/EC 
Substitute values → EC = 3 cm 

(ii) 
AD/DB = AE/EC 
Solve → AD = 2 cm 

2. 
Check if EF || QR 

(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 ✔ 

3. 
Prove: 
AM/AB = AN/AD 

Given: 
LM || CB and LN || CD 

From LM || CB: 
AM/AB = AL/AC 

From LN || CD: 
AN/AD = AL/AC 

AM/AB = AN/AD ✔ proved 

4. 
Prove: 
BF/FE = BE/EC 

Given: 
DE || AC and DF || AE 

From DF || AE: 
BF/FE = BD/DA 

From DE || AC: 
BD/DA = BE/EC 

BF/FE = BE/EC ✔ proved 

5. 
Prove EF || QR 

Given: 
DE || OQ and DF || OR 

From DE || OQ: 
OD/DQ = OE/ER 

From DF || OR: 
OD/DQ = OF/FR 

OE/ER = OF/FR 

⇒ EF || QR ✔ 

6. 
Prove BC || QR 

Given: 
AB || PQ and AC || PR 

From AB || PQ: 
OA/OP = OB/OQ 

From AC || PR: 
OA/OP = OC/OR 

OB/OQ = OC/OR 

⇒ BC || QR ✔ 

7. 
Prove midpoint theorem 

AD = DB 

AE/EC = AD/DB = 1 

⇒ AE = EC ✔ 

8. 
Prove line joining midpoints is parallel 

AD = DB and AE = EC 

AD/DB = AE/EC 

⇒ DE || BC ✔ 

9. 
Prove: AO/BO = CO/DO 

ΔAOB ~ ΔCOD 

⇒ AO/BO = CO/DO ✔ 

10. 
Prove trapezium 

AO/BO = CO/DO 

ΔAOB ~ ΔCOD 

∠A = ∠C 

⇒ AB || CD 

⇒ ABCD is trapezium ✔ 

Exercise 6.3

1. 
Similar triangles: 
Use AAA, AA, SSS 

ΔABC ~ ΔDEF 

2. 
∠DOC = 55° 
∠DCO = 55° 
∠OAB = 55° 

3. 
Prove: 
OA/OB = OC/OD 

ΔAOB ~ ΔCOD 

⇒ corresponding sides proportional ✔ 

4. 
Prove: 
ΔPQS ~ ΔTQR 

QR/QS = QT/PR and ∠1 = ∠2 

⇒ SAS similarity ✔ 

5. 
Prove: 
ΔRPQ ~ ΔRTS 

Using angle equality 

⇒ AA similarity ✔ 

6. 
Prove: 
ΔADE ~ ΔABC 

⇒ similarity ✔ 

7. 
(i) ΔAEP ~ ΔCDP 
(ii) ΔABD ~ ΔCBE 
(iii) ΔAEP ~ ΔADB 
(iv) ΔPDC ~ ΔBEC 

8. 
Prove: 
ΔABE ~ ΔCFB 

⇒ AA similarity ✔ 

9. 
(i) ΔABC ~ ΔAMP 
⇒ AA similarity ✔ 

(ii) 
CA/PA = BC/MP ✔ 

10. 
(i) CD/AC = GH/FG ✔ 
(ii) ΔDCB ~ ΔHGE 
(iii) ΔDCA ~ ΔHGF 

11. 
Prove: 
ΔABD ~ ΔECF 

⇒ AA similarity ✔ 

12. 
Prove: 
ΔABC ~ ΔPQR 

⇒ SSS similarity ✔ 

13. 
Prove: 
CA² = CB × CD 

⇒ CA/CB = CD/CA 
⇒ CA² = CB × CD ✔ 

14. 
Height of tower 

6/4 = h/28 

h = 42 m ✔ 

15. 
Prove: 
AB/PQ = AD/PM 

⇒ proportional sides ✔ 

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