Triangles in CBSE board papers
CBSE asked 174 different Triangles questions in the Class 10 Mathematics board papers of 2023, 2024, 2025 and 2026 (main and compartment exams, Standard and Basic courses). 18 of these questions have a worked solution on this page. Most of them test Section 6.4, Criteria for Similarity of Triangles (115 of 174).
Where Triangles questions came from, 2023–2026
A question printed in several question-paper sets counts once in each table. A question asked in two exams counts in both rows of the first table.
Questions by exam
| Year | Exam | Questions asked | Marks | Question types |
|---|---|---|---|---|
| 2026 | Main | 43 | 20 × 1 mark · 10 × 2 marks · 1 × 3 marks · 12 × 5 marks | 19 MCQs, 1 Assertion–Reason and 23 written answers |
| 2025 | Main | 55 | 18 × 1 mark · 18 × 2 marks · 19 × 5 marks | 17 MCQs, 1 Assertion–Reason and 37 written answers |
| 2025 | Compartment | 7 | 3 × 1 mark · 2 × 2 marks · 2 × 5 marks | 2 MCQs, 1 Assertion–Reason and 4 written answers |
| 2024 | Main | 44 | 15 × 1 mark · 10 × 2 marks · 1 × 3 marks · 2 × 4 marks · 16 × 5 marks | 13 MCQs, 2 Assertion–Reason, 2 case studies and 27 written answers |
| 2024 | Compartment | 9 | 3 × 1 mark · 4 × 2 marks · 2 × 5 marks | 3 MCQs and 6 written answers |
| 2023 | Main | 35 | 17 × 1 mark · 4 × 2 marks · 8 × 3 marks · 2 × 4 marks · 4 × 5 marks | 17 MCQs, 1 case study and 17 written answers |
| 2023 | Compartment | 10 | 2 × 1 mark · 1 × 2 marks · 4 × 3 marks · 3 × 5 marks | 2 MCQs and 8 written answers |
Questions by NCERT section and marks
| NCERT section | 1 mark | 2 marks | 3 marks | 4 marks | 5 marks | Total |
|---|---|---|---|---|---|---|
| 6.2 Similar Figures | 4 | 0 | 0 | 0 | 0 | 4 |
| 6.3 Similarity of Triangles | 20 | 12 | 5 | 0 | 17 | 54 |
| 6.4 Criteria for Similarity of Triangles | 46 | 32 | 8 | 4 | 25 | 115 |
| Mixed topics | 1 | 0 | 0 | 0 | 0 | 1 |
| All questions | 71 | 44 | 13 | 4 | 42 | 174 |
Most-tested sections
- 6.4 Criteria for Similarity of Triangles: 115 questions
- 6.3 Similarity of Triangles: 54 questions
- 6.2 Similar Figures: 4 questions
From 2023 to 2026, CBSE took most Triangles questions from Section 6.4 (115 of 174). These counts describe past papers; they are not predictions.
Topics in Triangles
Each NCERT section of the chapter, in book order: what CBSE asked from it and where to practise it.
6.2 Similar Figures
CBSE asked 4 questions from this section in the 2023–2026 papers (4 of 1 mark): 4 MCQs. 3 were printed in Standard papers and 1 in Basic papers.
| Marks | Questions asked | Years | Asked in Standard | Asked in Basic |
|---|---|---|---|---|
| 1 mark | 4 | 2025, 2026 | 3 | 1 |
Solved on this page: question 1, question 8, question 9, question 11.
6.3 Similarity of Triangles
Two triangles are similar exactly when their matching angles are equal and their matching sides are proportional. A line parallel to one side of a triangle, cutting the other two sides at distinct points, splits those sides in the same ratio. Conversely, a line that splits two sides of a triangle in the same ratio is parallel to the third side.
How to answer questions on this section
- First verify the line is parallel to one side before using Theorem 6.1.
- Start with AD/DB = AE/EC, invert to DB/AD = EC/AE, add 1 to get AB/AD = AC/AE, then invert to AD/AB = AE/AC.
- When equal ratios are given, apply Theorem 6.2 to prove parallelism.
- For trapezium problems, join a diagonal and apply Theorem 6.1 in each triangle.
- After showing parallel lines, use corresponding angles to connect angle and side facts.
Mistakes to avoid in this section
- Jumping straight to AD/AB = AE/AC without first writing AD/DB = AE/EC.
- Using Theorem 6.1 when the parallel condition is not given or proved.
- Applying Theorem 6.2 when the equal ratios are not on the two divided sides.
CBSE asked 54 questions from this section in the 2023–2026 papers (20 of 1 mark; 12 of 2 marks; 5 of 3 marks; 17 of 5 marks): 18 MCQs, 2 Assertion–Reason and 34 written answers. 30 were printed in Standard papers and 26 in Basic papers (2 in both).
| Marks | Questions asked | Years | Asked in Standard | Asked in Basic |
|---|---|---|---|---|
| 1 mark | 20 | 2023, 2024, 2025, 2026 | 12 | 8 |
| 2 marks | 12 | 2023, 2024, 2025, 2026 | 6 | 6 |
| 3 marks | 5 | 2023 | 1 | 4 |
| 5 marks | 17 | 2023, 2024, 2025, 2026 | 11 | 8 |
A question printed in both courses is counted in both columns (2 in both).
Solved on this page: question 3, question 4, question 13.
6.4 Criteria for Similarity of Triangles
If in two triangles corresponding angles are equal, then corresponding sides are in the same ratio, hence similar (AAA). If sides of one triangle are proportional to sides of another, corresponding angles are equal, hence similar (SSS). If one angle equals one angle and sides including these angles are proportional, triangles are similar (SAS). Two equal angles suffice by the angle-sum property, giving AA.
How to answer questions on this section
- Write similarity with matching vertices: if A maps to D, B to E, C to F, write ∆ABC ~ ∆DEF.
- In parallel-line figures, first mark alternate angles and vertically opposite angles; they give AA.
- For SSS, take ratios of corresponding sides in the same order, as Example 5 does.
- For SAS, confirm the equal angle lies between the two proportional sides, like OA/OC = OD/OB.
- After similarity, use corresponding angles or sides to find the missing measure, as in Example 5.
Mistakes to avoid in this section
- Writing ∆ABC ~ ∆EDF when the correct correspondence is A↔D, B↔E, C↔F; order must match.
- Using SAS when the equal angle is not the included angle between the proportional sides.
- Matching wrong sides in SSS, for example AB/DE with CA/EF instead of CA/FD.
CBSE asked 115 questions from this section in the 2023–2026 papers (46 of 1 mark; 32 of 2 marks; 8 of 3 marks; 4 of 4 marks; 25 of 5 marks): 44 MCQs, 2 Assertion–Reason, 3 case studies and 66 written answers. 67 were printed in Standard papers and 50 in Basic papers (2 in both).
| Marks | Questions asked | Years | Asked in Standard | Asked in Basic |
|---|---|---|---|---|
| 1 mark | 46 | 2023, 2024, 2025, 2026 | 26 | 20 |
| 2 marks | 32 | 2023, 2024, 2025, 2026 | 22 | 10 |
| 3 marks | 8 | 2023, 2024, 2026 | 3 | 5 |
| 4 marks | 4 | 2023, 2024 | 0 | 4 |
| 5 marks | 25 | 2023, 2024, 2025, 2026 | 16 | 11 |
A question printed in both courses is counted in both columns (2 in both).
Solved on this page: question 2, question 5, question 6, question 7, question 10, question 12, question 14, question 15, question 16, question 17, question 18.
1 more question uses several sections at once; the tables above count it under Mixed topics.
Key ideas used on this page
Similarity of Triangles
Similar figures
Corresponding angles of similar triangles
Corresponding sides of similar triangles are in the same ratio
Basic Proportionality Theorem
Similarity of polygons
Used in question 7
AA similarity criterion
Used in question 11
Converse of the Basic Proportionality Theorem
Used in question 13
6 solved questions last asked in 2026
Q1. Which types of triangles are always similar ?
- (A) Right-angled triangles
- (B) Acute-angled triangles
- (C) Isosceles triangles
- (D) Equilateral triangles
- Marks
- 1
- Level
- Basic
- Asked in
- 6 paper sets
- Years
- 2025, 2026
- Topic
- 6.2 Similar Figures
- Source
- (CBSE 2025 Mathematics Basic, Set 430/3/1, Q9, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(D) Equilateral triangles, because all equilateral triangles have equal corresponding angles and proportional corresponding sides.
Why? By the definition of similar figures, two polygons are similar when their corresponding angles are equal and their corresponding sides are in the same ratio; every equilateral triangle has all angles equal and all sides equal, so any two equilateral triangles satisfy this.
The full 1 mark is for the correct option.
Answer:
(D) Equilateral triangles
✓ Answer matches the CBSE answer key, 2025 Main, Paper 430/3/1: option (D).
Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Choosing isosceles triangles, because two isosceles triangles need not have their corresponding sides in the same ratio.
- Choosing right-angled triangles, because all right-angled triangles share only one 90° angle; the other two angles can differ.
- Choosing acute-angled triangles, because all angles being acute does not make the angles of the two triangles equal.
Q2. If \(\Delta ABC\) and \(\Delta DEF\) are similar such that \(2AB = DE\) and \(BC = 8\text{ cm}\) , then \(EF\) is equal to :
- (A) 4 cm
- (B) 8 cm
- (C) 12 cm
- (D) 16 cm
- Marks
- 1
- Level
- Standard
- Asked in
- 2 paper sets
- Years
- 2026
- Topic
- 6.4 Criteria for Similarity of Triangles
- Source
- (CBSE 2026 Mathematics Standard, Set 30/1/1, Q7, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(D) 16 cm, because ΔABC ~ ΔDEF gives DE/AB = EF/BC, and DE = 2AB, so EF = 2 × 8 = 16 cm.
Why? In similar triangles, corresponding sides are in the same ratio, so the side corresponding to BC is EF and it is multiplied by the same scale factor 2.
The full 1 mark is for the correct option.
Answer:
(D) 16 cm
✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/1/1: option (D).
Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Reading 2AB = DE as 'AB is twice DE' and choosing 4 cm; actually DE is twice AB, so the scale factor is 2 and EF is 16 cm.
- Matching BC with FD instead of EF; the order of vertices in ΔABC ~ ΔDEF fixes BC ↔ EF.
- Choosing 8 cm, thinking the triangles are congruent because only one side is given; similarity does not mean equal sides.
Q3. If in a \(\triangle ABC\), \(AB = 6\) cm and \(DE \parallel BC\) such that \(AE = \frac{1}{3} AC\), then the length of BD is
- (A) 2 cm
- (B) 3 cm
- (C) 4 cm
- (D) 5 cm
- Marks
- 1
- Level
- Standard
- Years
- 2026
- Topic
- 6.3 Similarity of Triangles
- Source
- (CBSE 2026 Mathematics Standard, Set 30(B), Q11, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(C) 4 cm, because DE ∥ BC, so by the Basic Proportionality Theorem, AD/DB = AE/EC = 1/2, giving AD = 1/3 × 6 = 2 cm and DB = 2/3 × 6 = 4 cm.
Why? The Basic Proportionality Theorem says a line parallel to one side divides the other two sides in the same ratio; this turns the given fraction AE/AC into the ratio AD/DB, from which BD follows.
The full 1 mark is for the correct option.
Answer:
(C) 4 cm
✓ Answer matches the CBSE answer key, 2026 Main, Paper 30(B): option (C).
Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Writing BD as 2 cm because AD is one-third of AB; the question asks for BD, the remaining 4 cm.
- Taking AE/EC as one-third, forgetting that EC is two-thirds of AC, so the ratio is 1/2.
- Assuming D is the midpoint of AB just because DE ∥ BC; the parallel line divides the sides in the ratio AE:EC, not equally.
Q4. ABCD is a trapezium in which AB || DC and E, F are points on AD and BC respectively such that EF || DC. If ED = 36 cm, BF = 70 cm and FC = 30 cm, then the length of AD is:
- (A) 124 cm
- (B) 120 cm
- (C) 110 cm
- (D) 114 cm
- Marks
- 1
- Level
- Standard
- Years
- 2026
- Topic
- 6.3 Similarity of Triangles
- Source
- (CBSE 2026 Mathematics Standard, Set 30(B), Q6, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(B) 120 cm, because AE/ED = BF/FC, so AE = 36 × 70/30 = 84 and AD = 84 + 36 = 120.
Why? In a trapezium with AB || DC and EF || AB, the non-parallel sides are divided in the same ratio: AE/ED = BF/FC. Then AD = AE + ED because E lies on AD.
The full 1 mark is for the correct option.
Answer:
(B) 120 cm
✓ Answer matches the CBSE answer key, 2026 Main, Paper 30(B): option (B).
Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Reversing the ratio as AE/ED = FC/BF, which gives AE = 36 × 30/70 = 108/7 cm, a wrong length.
- Adding ED + FC = 36 + 30 = 66 and calling it AD, even though FC is on side BC.
- Finding AE = 84 and stopping, without adding ED to get AD.
Q5. If \(\triangle DEF \sim \triangle PQR\) such that \(3DE = PQ\) and \(EF = 6\) cm, then the length of QR is:
- (A) 12 cm
- (B) 3 cm
- (C) 2 cm
- (D) 18 cm
- Marks
- 1
- Level
- Standard
- Years
- 2026
- Topic
- 6.4 Criteria for Similarity of Triangles
- Source
- (CBSE 2026 Mathematics Standard, Set 30(B), Q8, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(D) 18 cm, because 3DE = PQ, so PQ/DE = 3 and QR = 3 × EF = 3 × 6 = 18 cm.
Why? By the definition of similar triangles, corresponding sides are in the same ratio; from 3DE = PQ, every side of ΔPQR is 3 times the corresponding side of ΔDEF, so QR = 3 × EF.
The full 1 mark is for the correct option.
Answer:
(D) 18 cm
✓ Answer matches the CBSE answer key, 2026 Main, Paper 30(B): option (D).
Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Reading 3DE = PQ the wrong way round and taking PQ as one-third of DE, which gives 2 cm instead of 18 cm.
- Pairing EF with PR (or DE with QR) when writing the ratio of corresponding sides, so the given information cannot lead to QR.
- Dividing 6 cm by 3 because the scale factor was taken as 1/3; the scale factor is 3, so EF must be multiplied by 3.
Q6. It is given that \(\Delta ABC \sim \Delta QRP\) such that \(AB = 9\) cm, \(BC = 5\) cm and \(PR = 2\) cm. Length of side QR is :
- (A) 0.9 cm
- (B) \(\frac{5}{18}\) cm
- (C) \(\frac{10}{9}\) cm
- (D) 3.6 cm
- Marks
- 1
- Level
- Standard
- Years
- 2026
- Topic
- 6.4 Criteria for Similarity of Triangles
- Source
- (CBSE 2026 Mathematics Standard, Set 30/3/2, Q5, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(D) 3.6 cm, because AB/QR = BC/PR gives 9/QR = 5/2, so QR = 18/5 = 3.6 cm.
Why? In similar triangles, corresponding sides are in the same ratio; the order of vertices in the similarity symbol tells which sides correspond.
The full 1 mark is for the correct option.
Answer:
(D) 3.6 cm
✓ Answer matches the CBSE answer key, 2026 Main, Paper 30/3/2: option (D).
Open CBSE’s official marking scheme for the 2026 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Pairing BC with QR instead of PR, which leads to the wrong option 10/9 cm.
- Pairing PR with AB instead of BC; PR is the same side as RP and corresponds to BC, not to AB.
5 solved questions last asked in 2025
Q7. Which of the following is an incorrect statement ?
- (A) Two congruent triangles are also similar.
- (B) A square and a rhombus are not similar.
- (C) Two triangles are similar if their corresponding sides are proportional.
- (D) Two polygons are similar if and only if their corresponding sides are in proportion.
- Marks
- 1
- Level
- Basic
- Asked in
- 3 paper sets
- Compartment
- 2025
- Topic
- 6.4 Criteria for Similarity of Triangles
- Source
- (CBSE 2025 Compartment Mathematics Basic, Set 430/S/1, Q6, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(D) Two polygons are similar if and only if their corresponding sides are in proportion, because similarity of polygons also requires the corresponding angles to be equal.
Why? Similarity of polygons needs both equal corresponding angles and proportional corresponding sides; option D gives only the sides condition, so it is incorrect.
The full 1 mark is for the correct option.
Answer:
(D) Two polygons are similar if and only if their corresponding sides are in proportion.
✓ Answer matches the CBSE answer key, 2025 Compartment, Paper 430/S/1: option (D).
Open CBSE’s official marking scheme for the 2025 compartment exam to see how the board gives marks for this question.
Common mistakes to avoid
- Choosing C because you think proportional sides are not enough for triangles, but the SSS similarity criterion makes C true.
- Thinking a square and a rhombus are similar because all sides are in the same ratio, without checking that their corresponding angles are not equal.
- Selecting B as incorrect because a square and a rhombus have equal sides, forgetting that similarity also needs equal corresponding angles.
Q8. Which of the following statements is incorrect ?
- (A) Two congruent figures are always similar.
- (B) A square and a rhombus of the same area are always similar.
- (C) Two equilateral triangles are always similar.
- (D) Two similar triangles need not be congruent.
- Marks
- 1
- Level
- Standard
- Asked in
- 3 paper sets
- Years
- 2025
- Topic
- 6.2 Similar Figures
- Source
- (CBSE 2025 Mathematics Standard, Set 30/2/1, Q9, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(B) A square and a rhombus of the same area are always similar, because equal area does not force their corresponding angles to be equal, so they need not have the same shape.
Why? Two polygons are similar only when their corresponding angles are equal and their corresponding sides are in the same ratio; equal area alone guarantees neither.
The full 1 mark is for the correct option.
Answer:
(B) A square and a rhombus of the same area are always similar.
✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/2/1: option (B).
Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Choosing A because congruent figures have the same size, forgetting that same shape alone already makes two figures similar.
- Thinking that equal area means the same shape, so a square and a rhombus with equal area must be similar.
- Choosing D because similar triangles can be congruent, without reading the words 'need not'.
Q9. Which of the following statements is false ?
- (A) Two right triangles are always similar.
- (B) Two squares are always similar.
- (C) Two equilateral triangles are always similar.
- (D) Two circles are always similar.
- Marks
- 1
- Level
- Standard
- Asked in
- 3 paper sets
- Years
- 2025
- Topic
- 6.2 Similar Figures
- Source
- (CBSE 2025 Mathematics Standard, Set 30/5/1, Q8, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(A) Two right triangles are always similar., because two right triangles need not have equal corresponding angles, so they need not be similar.
Why? Two triangles are similar only when their corresponding angles are equal and their corresponding sides are in the same ratio; a right angle fixes only one angle, so two right triangles can have different shapes.
The full 1 mark is for the correct option.
Answer:
(A) Two right triangles are always similar.
✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/5/1: option (A).
Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Choosing B because two squares can be of different sizes, forgetting that similar figures may have different sizes.
- Thinking all right triangles are similar because each has a 90° angle, without checking the other two angles.
- Rejecting D because two circles of different radii are not congruent, confusing congruent with similar.
Q10. Given \(\triangle ABC \sim \triangle PQR\), \(\angle A = 30^\circ\) and \(\angle Q = 90^\circ\). The value of \((\angle R + \angle B)\) is
- (A) \(90^\circ\)
- (B) \(120^\circ\)
- (C) \(150^\circ\)
- (D) \(180^\circ\)
- Marks
- 1
- Level
- Standard
- Asked in
- 3 paper sets
- Years
- 2025
- Topic
- 6.4 Criteria for Similarity of Triangles
- Source
- (CBSE 2025 Mathematics Standard, Set 30/6/1, Q7, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(C) 150°, because ∠B = ∠Q = 90°, ∠C = 180° − 30° − 90° = 60° = ∠R, so ∠R + ∠B = 60° + 90° = 150°.
Why? Corresponding angles of similar triangles are equal, and the angle sum property of a triangle gives the third angle.
The full 1 mark is for the correct option.
Answer:
(C) 150°
✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/6/1: option (C).
Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Choosing 120° by adding the two given angles 30° and 90°, without finding ∠R.
- Taking ∠B as 30° because ∠A is 30°, instead of using the correspondence B ↔ Q, which makes ∠B equal to ∠Q (90°).
- Taking ∠R as 90° because Q is 90°, instead of using ∠R = ∠C = 60°.
Q11. In triangles ABC and DEF, ∠B = ∠E, ∠F = ∠C and AB = 3 DE. Then, the two triangles are :
- (A) congruent but not similar
- (B) congruent as well as similar
- (C) neither congruent nor similar
- (D) similar but not congruent
- Marks
- 1
- Level
- Standard
- Asked in
- 2 paper sets
- Years
- 2025
- Topic
- 6.2 Similar Figures
- Source
- (CBSE 2025 Mathematics Standard, Set 30/1/1, Q8, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(D) similar but not congruent, because ∠A = ∠D by the angle sum property, so ΔABC ~ ΔDEF by the AA similarity criterion, and AB/DE = 3/1, so corresponding sides are unequal.
Why? The AA similarity criterion requires only two pairs of equal angles; congruence additionally requires equal corresponding sides. A scale factor different from 1 allows similarity but blocks congruence.
The full 1 mark is for the correct option.
Answer:
(D) similar but not congruent
✓ Answer matches the CBSE answer key, 2025 Main, Paper 30/1/1: option (D).
Open CBSE’s official marking scheme for the 2025 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Choosing 'congruent as well as similar' after proving similarity, without checking that the corresponding sides are unequal.
- Choosing 'congruent but not similar' because the sides are unequal, forgetting that equal angles already prove similarity.
- Using the wrong correspondence, such as pairing AB with DF, and then wrongly concluding the triangles are neither similar nor congruent.
2 solved questions last asked in 2024
Q12. The perimeters of two similar triangles ABC and PQR are 56 cm and 48 cm respectively. PQ/AB is equal to
- (A) \(\frac{7}{8}\)
- (B) \(\frac{6}{7}\)
- (C) \(\frac{7}{6}\)
- (D) \(\frac{8}{7}\)
- Marks
- 1
- Level
- Standard
- Asked in
- 3 paper sets
- Years
- 2024
- Topic
- 6.4 Criteria for Similarity of Triangles
- Source
- (CBSE 2024 Mathematics Standard, Set 30/2/1, Q10, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(B) 6/7, because for similar triangles the ratio of the perimeters equals the ratio of the corresponding sides, so PQ/AB = 48/56 = 6/7.
Why? In similar triangles, every pair of corresponding sides is in the same ratio, and the perimeters are also in that same ratio.
The full 1 mark is for the correct option.
Answer:
(B) 6/7
✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/2/1: option (B).
Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Choosing 7/6, which is 56/48, the ratio of the larger perimeter to the smaller one, instead of the asked PQ/AB = 48/56.
- Choosing 8/7 by taking the reciprocal of 7/8 without checking which ratio the question asks for.
- Leaving the answer as 48/56 without reducing it to 6/7, so it does not match any option.
Q13. If the diagonals of a quadrilateral divide each other proportionally, then it is a :
- (A) parallelogram
- (B) rectangle
- (C) square
- (D) trapezium
- Marks
- 1
- Level
- Standard
- Asked in
- 3 paper sets
- Years
- 2024
- Topic
- 6.3 Similarity of Triangles
- Source
- (CBSE 2024 Mathematics Standard, Set 30/3/1, Q11, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(D) trapezium, because if the diagonals divide each other in the same ratio, then one pair of opposite sides is parallel by the converse of the Basic Proportionality Theorem.
Why? The converse of the Basic Proportionality Theorem says that if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side. Applying this to the divided diagonals gives AB ∥ DC, so the quadrilateral is a trapezium.
The full 1 mark is for the correct option.
Answer:
(D) trapezium
✓ Answer matches the CBSE answer key, 2024 Main, Paper 30/3/1: option (D).
Open CBSE’s official marking scheme for the 2024 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Choosing parallelogram because a parallelogram's diagonals bisect each other; the given condition only guarantees one pair of opposite sides parallel, not both pairs.
- Choosing rectangle or square because the ratio can be 1 : 1; the condition gives no information about angles or equal sides.
5 solved questions last asked in 2023
Q14. The sides of two similar triangles are in the ratio 4 : 7. The ratio of their perimeters is
- (A) 4 : 7
- (B) 12 : 21
- (C) 16 : 49
- (D) 7 : 4
- Marks
- 1
- Level
- Basic
- Asked in
- 3 paper sets
- Years
- 2023
- Topic
- 6.4 Criteria for Similarity of Triangles
- Source
- (CBSE 2023 Mathematics Basic, Set 430/1/1, Q8, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(A) 4 : 7, because the ratio of the perimeters of similar triangles is the same as the ratio of their corresponding sides.
Why? For similar triangles, the corresponding sides are in the same ratio; the perimeter is the sum of the sides, so it is in that same ratio.
The full 1 mark is for the correct option.
Answer:
(A) 4 : 7
✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/1/1: option (A).
Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Choosing 16 : 49, because it is the square of 4 : 7, but perimeters are not squared.
- Choosing 7 : 4, because the order of the given ratio was reversed.
Q15. If \(\triangle ABC \sim \triangle PQR\) with \(\angle A = 32^\circ\) and \(\angle R = 65^\circ\) , then the measure of \(\angle B\) is :
- (A) \(32^\circ\)
- (B) \(65^\circ\)
- (C) \(83^\circ\)
- (D) \(97^\circ\)
- Marks
- 1
- Level
- Standard
- Asked in
- 3 paper sets
- Years
- 2023
- Topic
- 6.4 Criteria for Similarity of Triangles
- Source
- (CBSE 2023 Mathematics Standard, Set 30/2/1, Q3, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(C) 83°, because ∠C = ∠R = 65° and ∠B = 180° − 32° − 65° = 83°.
Why? Corresponding angles of similar triangles are equal, and the angle sum property of a triangle gives the third angle.
The full 1 mark is for the correct option.
Answer:
(C) 83°
✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/2/1: option (C).
Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Choosing 32° because it is the only given angle of △ABC, without using the correspondence C ↔ R.
- Choosing 65° by pairing ∠B with ∠R, forgetting that B corresponds to Q and C corresponds to R.
- Choosing 97° by adding 32° and 65° without subtracting the sum from 180°.
Q16. If ΔABC ~ ΔDEF and ∠A = 47°, ∠E = 83°, then ∠C is equal :
- (A) 47°
- (B) 50°
- (C) 83°
- (D) 130°
- Marks
- 1
- Level
- Basic
- Asked in
- 2 paper sets
- Years
- 2023
- Topic
- 6.4 Criteria for Similarity of Triangles
- Source
- (CBSE 2023 Mathematics Basic, Set 430/4/2, Q16, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(B) 50°, because ∠B = ∠E = 83° for similar triangles, and ∠C = 180° − 47° − 83° = 50°.
Why? Corresponding angles of similar triangles are equal, so ∠B = ∠E; the angle sum property of a triangle then fixes the third angle.
The full 1 mark is for the correct option.
Answer:
(B) 50°
✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/4/2: option (B).
Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Choosing 83° for ∠C because the given angle E is 83°, without using the order of letters in ΔABC ~ ΔDEF, which pairs C with F, not E.
- Choosing 130°, the sum 47° + 83°, instead of subtracting this sum from 180°.
- Using the wrong correspondence, such as pairing A with E, although ΔABC ~ ΔDEF fixes A with D, B with E and C with F.
Q17. If \(\Delta ABC \sim \Delta DEF\) and \(\angle A = 47^\circ\), \(\angle E = 83^\circ\), then \(\angle C\) is equal :
- (A) \(47^\circ\)
- (B) \(50^\circ\)
- (C) \(83^\circ\)
- (D) \(130^\circ\)
- Marks
- 1
- Level
- Basic
- Years
- 2023
- Topic
- 6.4 Criteria for Similarity of Triangles
- Source
- (CBSE 2023 Mathematics Basic, Set 430/4/1, Q8, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(B) 50°, because ∠B = ∠E = 83° and ∠C = 180° − 47° − 83° = 50°.
Why? Corresponding angles of similar triangles are equal, and the three angles of a triangle add up to 180°.
The full 1 mark is for the correct option.
Answer:
(B) 50°
✓ Answer matches the CBSE answer key, 2023 Main, Paper 430/4/1: option (B).
Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Choosing 130° because 47° and 83° were added, forgetting that the angle sum needs 180°.
- Choosing 83° because ∠E is given, forgetting that ∠C corresponds to ∠F, not ∠E.
- Choosing 47° because ∠A is given, confusing ∠C with ∠A.
Q18. If \(\triangle PQR \sim \triangle ABC\); \(PQ = 6\) cm, \(AB = 8\) cm and the perimeter of \(\triangle ABC\) is 36 cm, then the perimeter of \(\triangle PQR\) is
- (A) 20.25 cm
- (B) 27 cm
- (C) 48 cm
- (D) 64 cm
- Marks
- 1
- Level
- Standard
- Years
- 2023
- Topic
- 6.4 Criteria for Similarity of Triangles
- Source
- (CBSE 2023 Mathematics Standard, Set 30/6/3, Q12, 1 mark)
- Status
- Verified solution
Step-by-step solution
Step 1: Choose the option
1 mark(B) 27 cm, because the ratio of perimeters of similar triangles equals the ratio of corresponding sides, PQ/AB = 6/8, so perimeter of ΔPQR = 6/8 × 36 = 27 cm.
Why? For similar triangles, the ratio of any two corresponding sides is the same, and the perimeters are in the same ratio.
The full 1 mark is for the correct option.
Answer:
(B) 27 cm
✓ Answer matches the CBSE answer key, 2023 Main, Paper 30/6/3: option (B).
Open CBSE’s official marking scheme for the 2023 main exam to see how the board gives marks for this question.
Common mistakes to avoid
- Reversing the correspondence: taking AB/PQ as 8/6 and multiplying 36 by it, which gives 48 cm (option C).
- Subtracting the difference of the given sides, 8 − 6 = 2, from 36, which gives 34 cm.
- Squaring the scale factor: 36 × (6/8)² = 20.25 cm (option A).
Most repeated Triangles questions
The questions CBSE printed in the most question-paper sets, 2023–2026. Each one is solved where the line points.
- Printed in 6 question-paper sets (2025 and 2026) · 1 mark · Basic · question 1 on this page
- Printed in 3 question-paper sets (2023) · 1 mark · Basic · question 14 on this page
- Printed in 3 question-paper sets (2023) · 1 mark · Standard · question 15 on this page
- Printed in 3 question-paper sets (2024) · 1 mark · Standard · question 12 on this page
- Printed in 3 question-paper sets (2024) · 1 mark · Standard · question 13 on this page
Official papers behind this chapter
Triangles was asked in 117 of the 120 question-paper sets of 2023–2026. The table names the official CBSE files for each exam in which it was asked.
The board publishes the question papers on cbse.gov.in and the marking schemes on cbse.gov.in, where you can see how marks are given.
| Year | Exam | Course | Sets with a Triangles question | Question paper | Marking scheme |
|---|---|---|---|---|---|
| 2026 | Main | Standard (041) | 10 of 10 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2026 main exam (one ZIP file of about 61.9 MB, Class 10 Maths Standard and Basic together) to read the full paper. | Open CBSE's official marking scheme for the 2026 main exam (one ZIP file of about 22.8 MB, Class 10 Maths Standard (041)) to see how marks are given. |
| 2026 | Main | Basic (241) | 10 of 10 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2026 main exam (one ZIP file of about 61.9 MB, Class 10 Maths Standard and Basic together) to read the full paper. | Open CBSE's official marking scheme for the 2026 main exam (one ZIP file of about 29.1 MB, Class 10 Maths Basic (241)) to see how marks are given. |
| 2025 | Main | Standard (041) | 17 of 17 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 54.5 MB, Class 10 Maths Standard (041)) to read the full paper. | Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given. |
| 2025 | Main | Basic (241) | 10 of 12 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2025 main exam (one ZIP file of about 49.9 MB, Class 10 Maths Basic (241)) to read the full paper. | Open CBSE's official marking scheme for the 2025 main exam (one ZIP file of about 56.9 MB, Class 10 Maths Standard and Basic together) to see how marks are given. |
| 2025 | Compartment | Standard (041) | 3 of 3 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 3.4 MB, Class 10 Maths Standard (041)) to read the full paper. | Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 8.1 MB, Class 10 Maths Standard (041)) to see how marks are given. |
| 2025 | Compartment | Basic (241) | 3 of 3 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2025 compartment exam (one ZIP file of about 2.4 MB, Class 10 Maths Basic (241)) to read the full paper. | Open CBSE's official marking scheme for the 2025 compartment exam (one ZIP file of about 18.7 MB, Class 10 Maths Basic (241)) to see how marks are given. |
| 2024 | Main | Standard (041) | 14 of 14 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 35.3 MB, Class 10 Maths Standard (041)) to read the full paper. | Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.4 MB, Class 10 Maths Standard (041)) to see how marks are given. |
| 2024 | Main | Basic (241) | 14 of 14 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2024 main exam (one ZIP file of about 43.4 MB, Class 10 Maths Basic (241)) to read the full paper. | Open CBSE's official marking scheme for the 2024 main exam (one ZIP file of about 23.8 MB, Class 10 Maths Basic (241)) to see how marks are given. |
| 2024 | Compartment | Standard (041) | 3 of 3 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2024 compartment exam (one ZIP file of about 3.2 MB, Class 10 Maths Standard (041)) to read the full paper. | Open CBSE's official marking scheme for the 2024 compartment exam (one ZIP file of about 7.1 MB, Class 10 Maths Standard (041)) to see how marks are given. |
| 2024 | Compartment | Basic (241) | 4 of 4 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2024 compartment exam (one ZIP file of about 2.2 MB, Class 10 Maths Basic (241)) to read the full paper. | Open CBSE's official marking scheme for the 2024 compartment exam (one ZIP file of about 5.6 MB, Class 10 Maths Basic (241)) to see how marks are given. |
| 2023 | Main | Standard (041) | 12 of 12 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 44 MB, Class 10 Maths Standard (041)) to read the full paper. | Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 17.4 MB, Class 10 Maths Standard (041)) to see how marks are given. |
| 2023 | Main | Basic (241) | 11 of 12 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2023 main exam (one ZIP file of about 49.4 MB, Class 10 Maths Basic (241)) to read the full paper. | Open CBSE's official marking scheme for the 2023 main exam (one ZIP file of about 26 MB, Class 10 Maths Basic (241)) to see how marks are given. |
| 2023 | Compartment | Standard (041) | 3 of 3 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2023 compartment exam (one ZIP file of about 7.6 MB, Class 10 Maths Standard (041)) to read the full paper. | Open CBSE's official marking scheme for the 2023 compartment exam (one ZIP file of about 3.3 MB, Class 10 Maths Standard (041)) to see how marks are given. |
| 2023 | Compartment | Basic (241) | 3 of 3 question-paper sets had a Triangles question | Open CBSE's official question paper for the 2023 compartment exam (one ZIP file of about 7.2 MB, Class 10 Maths Basic (241)) to read the full paper. | Open CBSE's official marking scheme for the 2023 compartment exam (one ZIP file of about 7.5 MB, Class 10 Maths Basic (241)) to see how marks are given. |
Continue studying
To revise the lessons behind these questions, read NCERT Chapter 6 (official PDF), the chapter exactly as NCERT publishes it on ncert.nic.in.
Sources and data verification
The counts on this page come from 120 CBSE Class 10 Mathematics question-paper sets: 62 Standard (041) and 58 Basic (241), from the main exams of 2023, 2024, 2025 and 2026 and the compartment exams of 2023, 2024 and 2025. A question printed in several sets counts once; these counts describe past papers and are not predictions.
CBSE settles the question papers and the marking schemes: each question is reproduced as printed in its CBSE paper, with its year, paper code and question number, and each answer shown here matches the official marking scheme or answer key. NCERT settles the textbook: chapter and section names follow NCERT Mathematics Class X (reprint 2026-27).
Every solution is original. The mark beside each step follows the official CBSE marking scheme where the scheme shows it; a mark labelled “(suggested)” follows the board’s usual pattern for that type of question.
The official question-paper and marking-scheme links on this page were checked and open on 9 October 2026.
Reference: CBSE Class 10 Mathematics board question papers via cbse.gov.in; all solutions are original. learncbse.net is an independent website and is not affiliated with CBSE or NCERT.