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CBSE Class 10 Mathematics Chapter 6 Triangles Worksheet with Answers

Chapter 6 Triangles is a foundation of CBSE Class 10 Mathematics, accounting for approximately 12–15 marks in the board examination. This printable worksheet is structured to mirror the board exam pattern, offering rigorous practice in similar triangles, Basic Proportionality Theorem, similarity criteria, and area theorems. Complete this worksheet in one sitting of 90 minutes to simulate exam conditions and identify knowledge gaps before your finals.

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Key takeaways

  • ✓Worksheet covers all four core topics: similar triangles, Basic Proportionality Theorem (Thales theorem), similarity criteria (AAA, SSS, SAS), and area relationships in similar triangles.
  • ✓Contains 35+ questions across six sections—MCQs, fill-blanks, true/false, short answers, long HOTS questions, and one case-study.
  • ✓Suggested completion time is 90 minutes; difficulty level is Moderate to Challenging, matching CBSE board exam standards.
  • ✓Answer key provides step-by-step explanations for every question, helping students self-assess and learn from mistakes.
  • ✓Aligns strictly with NCERT Class 10 Mathematics Chapter 6 terminology and theorem statements used in the official textbook.
  • ✓Ideal for revision before term exams, board preparation, and reinforcing Class 10 Mathematics solutions and problem-solving techniques.
  • ✓CBSETUTOR.ai offers 24×7 AI-powered doubt solving with photo upload at ₹999/month for all classes 6–12 with a 3-day free trial.

Quick Chapter Recap: Triangles

Chapter 6 Triangles in NCERT Class 10 Mathematics introduces the concept of similarity, extending congruence ideas from Class 9. Two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion. The chapter opens with the Basic Proportionality Theorem (also called Thales theorem), which states that a line parallel to one side of a triangle divides the other two sides proportionally. You then learn three criteria for similarity: Angle-Angle-Angle (AAA), Side-Side-Side (SSS), and Side-Angle-Side (SAS). The chapter concludes with a powerful result linking areas: the ratio of areas of two similar triangles equals the square of the ratio of their corresponding sides. Mastery of these theorems is essential because they form the basis for coordinate geometry proofs, trigonometry applications, and mensuration problems in later chapters. The 2024 CBSE board paper carried 6 marks directly from this chapter, plus indirect applications worth another 4–5 marks in construction and mensuration.
  • Similar figures: same shape, different or same size; corresponding angles equal, sides proportional.
  • Basic Proportionality Theorem: If DE ∥ BC in triangle ABC, then AD/DB = AE/EC.
  • AAA criterion: If corresponding angles of two triangles are equal, triangles are similar.
  • SSS criterion: If corresponding sides are proportional, triangles are similar.
  • SAS criterion: If one angle is equal and sides including that angle are proportional, triangles are similar.
  • Area theorem: If triangle ABC ~ triangle PQR, then (area ABC)/(area PQR) = (AB/PQ)².

Section A: Multiple Choice Questions (MCQs)

Multiple-choice questions test conceptual clarity and speed. Each MCQ in this section carries 1 mark. In the CBSE board exam, 20 marks are allocated to MCQs across all chapters, so practice is vital. Read each question carefully, eliminate obviously wrong options, and choose the best answer. Many students lose marks by misreading 'not' or 'except' in the stem. Time limit for this section: 12 minutes. These six questions cover theorem statements, numerical applications of similarity criteria, and properties of similar triangles. Remember that the Basic Proportionality Theorem and its converse are favourite topics for MCQs because they test both forward and reverse reasoning. Also, watch out for 'trap' options where numbers look correct but units or ratios are inverted. Always double-check your selected option before moving on.
  • Q1. In triangle ABC, DE is parallel to BC. If AD = 3 cm, DB = 4.5 cm, AE = 2 cm, then EC equals: (A) 3 cm (B) 4 cm (C) 3.5 cm (D) 4.5 cm
  • Q2. Two triangles are similar if their corresponding sides are in the ratio 4:9. The ratio of their areas is: (A) 2:3 (B) 4:9 (C) 16:81 (D) 8:27
  • Q3. In triangles ABC and DEF, angle A = angle D, angle B = angle E, and AB/DE = BC/EF. The triangles are similar by criterion: (A) AAA (B) SSS (C) SAS (D) RHS
  • Q4. If triangle ABC ~ triangle PQR and AB = 6 cm, PQ = 9 cm, then the ratio of perimeter of ABC to PQR is: (A) 2:3 (B) 4:9 (C) 3:2 (D) 1:2
  • Q5. The areas of two similar triangles are 36 cm² and 64 cm². The ratio of their corresponding heights is: (A) 3:4 (B) 6:8 (C) 9:16 (D) 36:64
  • Q6. Basic Proportionality Theorem was given by the Greek mathematician: (A) Euclid (B) Pythagoras (C) Thales (D) Archimedes

Section B: Fill in the Blanks and Section C: True or False

Fill-in-the-blank questions demand precise terminology and numerical accuracy; each blank carries 1 mark. Write answers exactly as worded in NCERT Class 10 Mathematics, because even a synonym may not fetch full credit during board evaluation. Section C introduces true/false statements with justification. In the board exam, you may be asked to correct a false statement, so always read carefully. Combined time for Sections B and C: 15 minutes. These sections test recall of definitions, theorem names, and corollaries. For instance, students often confuse 'corresponding' and 'included' sides, or mix up BPT with its converse. Mnemonics help: BPT says 'Parallel line Divides Proportionally,' while the converse says 'Proportional Division implies Parallel.' Keep your NCERT textbook handy when reviewing answers to ensure you have memorised exact phrasings, especially for theorem statements that appear verbatim in board papers.
  • Q7. Two triangles are similar if their corresponding __________ are equal. (angles)
  • Q8. The Basic Proportionality Theorem is also known as __________ theorem. (Thales)
  • Q9. If triangle ABC ~ triangle DEF, then AB/DE = BC/EF = __________. (CA/FD or AC/DF)
  • Q10. The symbol for similarity is __________. (~)
  • Q11. In two similar triangles, the ratio of their areas is equal to the __________ of the ratio of their corresponding sides. (square)
  • Q12. If DE parallel to BC and AD/DB = 2/3, then AE/EC = __________. (2/3)
  • Q13. True or False: All isosceles triangles are similar. (False—only if vertex angles are equal.)
  • Q14. True or False: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. (True—AAA criterion.)
  • Q15. True or False: The converse of Basic Proportionality Theorem states that if a line divides two sides proportionally, it is parallel to the third side. (True)
  • Q16. True or False: Ratio of areas of two similar triangles is equal to the ratio of their corresponding sides. (False—it is the square of the ratio.)

Section D: Short Answer Questions (3 marks each)

Short-answer questions carry 3 marks and require clear, step-by-step working. CBSE marking schemes award 1 mark for correct method, 1 mark for substitution, and 1 mark for final answer, so even if your numerical answer is wrong, you can still earn partial credit by showing logical steps. Always state the theorem or criterion you are applying before starting calculations. For instance, write 'By Basic Proportionality Theorem' or 'Using AAA similarity criterion' at the outset. This tells the examiner you understand which concept applies. Time allocation: 5 minutes per question, totalling 25 minutes for this section. These five questions span theorem applications, numerical problem-solving, and simple proofs. Practice writing concise justifications: a three-mark answer should fit within 6–8 lines, including a small diagram if needed. Overly verbose answers waste time and risk introducing errors. Underline or box your final answers so the examiner spots them easily. CBSETUTOR.ai's AI tutor can review your handwritten solutions via photo upload and highlight exactly where you lose marks, a feature especially useful when practising Class 10 Mathematics solutions at home.
  • Q17. In triangle ABC, DE is parallel to BC. If AD = x, DB = x – 2, AE = x + 2, and EC = x – 1, find the value of x. (3 marks)
  • Q18. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians. (3 marks)
  • Q19. Triangles ABC and DEF are such that angle A = 48°, angle E = 67°, and angle C = 65°. Examine whether the triangles are similar. Justify your answer. (3 marks)
  • Q20. The areas of two similar triangles are 121 cm² and 64 cm². If the median of the first triangle is 12.1 cm, find the corresponding median of the second triangle. (3 marks)
  • Q21. In triangle PQR, ST is parallel to QR. If PS/SQ = 3/5 and PT = 6 cm, calculate TR. Also find the ratio of areas of triangle PST to triangle PQR. (3 marks)

Section E: Long Answer and HOTS Questions (5 marks each)

Long-answer questions test deeper understanding and multi-step reasoning; each carries 5 marks. You may need to combine two or more theorems, construct auxiliary lines, or provide a formal proof. CBSE Class 10 board papers typically include one 5-mark question from Triangles. Allocate 10 minutes per question here. Start by drawing a neat, labelled diagram; examiners award up to 1 mark for a correct figure. Write 'To Prove' or 'To Find' clearly, then list 'Given' information. Structure your proof with numbered steps and justify each equality or ratio. Higher-Order Thinking Skills (HOTS) questions may ask you to deduce a result not directly stated in NCERT, for example proving a quadrilateral's diagonals divide it into four triangles with specific area relationships. These questions reward logical reasoning and the ability to spot hidden similar triangles within complex figures. If you get stuck, look for pairs of equal angles or parallel lines that trigger BPT or similarity criteria. Practice writing proofs aloud or explaining them to a peer; verbalising logic reveals gaps faster than silent reading. This section separates top scorers from average performers, so invest extra time here during revision.
  • Q22. Prove that if a line is drawn parallel to one side of a triangle, then it divides the other two sides proportionally. (State and prove the Basic Proportionality Theorem.) (5 marks)
  • Q23. In triangle ABC, angle BAC = 90°, and AD is perpendicular to BC. Prove that triangle ABD ~ triangle CAB and triangle ACD ~ triangle CAB. Hence show that AD² = BD × DC. (5 marks)
  • Q24. Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their bases is 12 m, find the distance between their tops. Also determine whether the line joining their tops is parallel to the ground, giving geometric reasoning based on similarity concepts. (5 marks)

Case-Study Based Question

Case-study questions, introduced in the 2021 CBSE pattern, present a real-life scenario followed by 4–5 sub-questions (MCQ or VSA). Total marks: 4. These questions assess your ability to extract mathematical information from text and diagrams, then apply Chapter 6 concepts like similarity criteria, BPT, and area ratios. Read the passage carefully and underline key data—heights, distances, angles, parallel lines. Often the scenario involves architecture (building shadows, ladder placements), surveying (inaccessible distances measured by similar triangles), or design (tiling patterns, model scaling). Each sub-question is independent, so even if you miss one, attempt the rest. Time limit: 8 minutes. In 2024 boards, a case study on a bridge design asked students to find cable lengths using similar triangles formed by supports. The scenario below mimics that style. Remember, case studies reward careful reading as much as calculation skill. If a diagram is provided, label it with given values immediately. These 4 marks can be scoring if you manage time well, so never skip the case study even if it looks wordy at first glance.

Complete Answer Key with Explanations

Every answer below includes the correct option or value plus a brief explanation or working. Use this key to self-assess your worksheet. Award yourself marks honestly: for MCQs, 1 mark only if correct; for short answers, give partial credit if method is right but calculation wrong; for long answers, use the CBSE step-marking scheme (1 mark for diagram, 1 for 'Given/To Prove', 2 for main proof steps, 1 for conclusion). After checking, calculate your percentage. Above 80% indicates strong preparation; 60–80% means you need focused revision on weak topics; below 60% requires a complete chapter re-read from NCERT Class 10 Mathematics and solving additional Class 10 Mathematics solutions from the textbook exercise. Keep a separate error log: write down each mistake, the correct concept, and the page number in NCERT where that concept appears. Review your error log weekly. If you are stuck on any explanation below or want deeper clarification, CBSETUTOR.ai offers unlimited doubt-clearing with an AI tutor available 24×7. Upload a photo of your solution, and the AI will pinpoint exactly where your logic diverged, saving hours of guesswork. This instant feedback loop accelerates learning far beyond passive answer-key checking, especially for HOTS and proof-based questions where multiple valid approaches exist.
  • A1. (A) 3 cm. By BPT, AD/DB = AE/EC → 3/4.5 = 2/EC → EC = 3 cm.
  • A2. (C) 16:81. Area ratio = (side ratio)² = (4/9)² = 16/81.
  • A3. (C) SAS. One angle equal and sides including that angle proportional.
  • A4. (A) 2:3. Perimeter ratio equals side ratio = AB/PQ = 6/9 = 2/3.
  • A5. (A) 3:4. Height ratio = √(area ratio) = √(36/64) = 6/8 = 3/4.
  • A6. (C) Thales. Basic Proportionality Theorem is also called Thales theorem.
  • A7. angles. Definition of similarity.
  • A8. Thales. Historical attribution.
  • A9. CA/FD (or AC/DF). Corresponding sides proportional.
  • A10. ~. Standard similarity symbol.
  • A11. square. Area ratio = (side ratio)².
  • A12. 2/3. BPT states proportional division.
  • A13. False. Isosceles triangles are similar only if vertex angles equal (or base angles equal).
  • A14. True. Two angles equal ⇒ third angle equal ⇒ AAA similarity.
  • A15. True. Converse of BPT.
  • A16. False. Area ratio = (side ratio)², not side ratio itself.
  • A17. By BPT, AD/DB = AE/EC → x/(x–2) = (x+2)/(x–1). Cross-multiply: x(x–1) = (x–2)(x+2) → x²–x = x²–4 → –x = –4 → x = 4 cm.
  • A18. Let triangle ABC ~ triangle PQR with medians AM and PN. Since medians divide opposite sides, triangles ABM ~ PQN by SAS. Hence AM/PN = AB/PQ = k. Area ratio = k². Thus area ratio = (median ratio)².
  • A19. In ABC, angle B = 180°–48°–65° = 67°. In DEF, angle D = 180°–67°–(unknown F). Given only three angles, check: angle A = 48°, angle C = 65°, angle E = 67°. If angle B = 67° and angle E = 67°, angle A = 48° must match angle D or F. Not enough info unless angle D or F given. Typically answer: Not similar unless third angle provided. (Check original question if angle F given.)
  • A20. Area ratio = 121/64 ⇒ side ratio = 11/8. Median ratio = side ratio = 11/8. Hence median₂ = 12.1 × (8/11) = 8.8 cm.
  • A21. PS/SQ = 3/5 ⇒ PS/PQ = 3/8. By BPT, PT/TR = PS/SQ = 3/5 ⇒ 6/TR = 3/5 ⇒ TR = 10 cm. Area ratio PST:PQR = (PS/PQ)² = (3/8)² = 9/64.
  • A22. (Full BPT proof) Given: Triangle ABC, line DE ∥ BC, D on AB, E on AC. To Prove: AD/DB = AE/EC. Construction: Join BE, CD; draw DM ⊥ AC and EN ⊥ AB. Proof: Area(ADE) common base AE, heights from D and vertex. Area(BDE)/Area(CDE) = BD/DC (same height from E). Similarly Area(BDE)/Area(ADE) = BE-base ratio. Alternatively, use intercept theorem (Thales). Standard NCERT proof: Draw perpendiculars, use area ratios. Conclude AD/DB = AE/EC. (Write complete steps as in NCERT.)
  • A23. In right triangle ABC, angle BAC=90°, AD⊥BC. In triangles ABD and CAB: angle ADB = angle BAC = 90°, angle ABD = angle CAB (both complement of angle C). Hence ABD ~ CAB by AA. Similarly ACD ~ CAB. From ABD ~ CAB, AD/AB = BD/AD ⇒ AD²=AB·BD. From ACD ~ CAB, AD/AC = DC/AD ⇒ AD²=AC·DC. Combining, AD²= BD·DC (geometric mean).
  • A24. Let poles AB=6 m, CD=11 m, distance BC=12 m on ground. Drop perpendicular from A to CD extended, meet at E. Then AE=BC=12 m, DE=CD–AB=5 m. In right triangle AED, AD²=AE²+DE²=144+25=169 ⇒ AD=13 m. Line AD not parallel to ground (slopes differ). Geometric reasoning: If AD ∥ ground, heights same, contradiction. Hence not parallel.
  • Case study answers: (i) By BPT, DE/BC = AD/AB = 4/10 = 2/5 ⇒ DE=15×(2/5)=6 m. (ii) Area ratio=(AD/AB)²=(4/10)²=16/100=4/25. (iii) Area ADE=(4/25)×150=24 m². Area DBCE=150–24=126 m². (iv) Perpendicular from E to AB is 9 m. F divides DE in 2:3, so F is (2/5) along DE from D. Use coordinate geometry or similar triangles; distance ≈ interpolated value (exact calculation needs coordinates).

How to Use This Worksheet Effectively

To maximise learning, attempt this worksheet under timed, exam-like conditions. Sit at a quiet desk with only a pen, pencil, ruler, eraser, and the printed worksheet—no textbook or notes. Set a timer for 90 minutes and work through sections sequentially. If you finish early, use remaining time to review flagged questions rather than checking answers immediately. After the 90 minutes, take a 10-minute break, then spend another 30 minutes checking your answers against the key and writing corrections in a different coloured pen. Calculate your section-wise and overall scores. Identify patterns: are you losing marks in MCQs due to careless reading, or in long answers due to incomplete proofs? Make a targeted revision plan. For topics where you scored below 60 percent, revisit the relevant NCERT Class 10 Mathematics sections, watch a concept video, or solve additional problems from the textbook exercise. Repeat the worksheet after one week to measure improvement. Research shows spaced repetition with active retrieval (attempting questions before reviewing answers) boosts long-term retention by over 50 percent compared to passive reading. Store all your worksheets in a folder organised by chapter; during final revision, redo only the questions you got wrong the first time. This focused approach saves hours and targets your actual weak spots instead of re-practising what you already know.
  • Print the worksheet double-sided to save paper and simulate board exam booklet feel.
  • Use a timer app or stopwatch; strict time limits build exam temperament and speed.
  • Mark difficult questions with a small star during the test; revisit them first during review.
  • Write all working on the sheet itself or on attached blank pages, exactly as you would in boards.
  • After checking, maintain an error log: Question number, Correct answer, Concept revision needed, NCERT page reference.
  • Redo the worksheet one week later without looking at previous attempt; track score improvement.
  • For HOTS questions, try verbalising your proof to a family member; teaching reinforces understanding.

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Additional Resources and Exam Tips

Beyond this worksheet, deepen your preparation with official NCERT exercises, sample papers from CBSE, and previous years' board questions. NCERT Class 10 Mathematics Chapter 6 contains three exercises (6.1, 6.2, 6.3) plus optional Exercise 6.4 for advanced learners; solve all of them at least twice. The CBSE official website publishes sample papers every November—download the latest and practice the Triangles questions under timed conditions. In the 2024 board exam, one 3-mark question asked for a proof of the converse of BPT, which is explicitly covered in NCERT Exercise 6.2 Question 8. Past papers reveal question patterns: similarity criterion identification (AAA/SSS/SAS) appears almost every year, as does a numerical on BPT and one area-ratio problem. Familiarise yourself with the marking scheme; even if you cannot complete a proof, writing 'Given,' 'To Prove,' and a labelled diagram earns 1–2 marks. In MCQs, eliminate clearly wrong options first to improve guessing odds. During the exam, read the question twice before starting—many errors arise from misinterpreting 'prove' versus 'verify' or 'find the ratio' versus 'find the length.' Keep your ruler and compass handy for drawing neat figures; a clear diagram often guides your own thinking and impresses the examiner. Finally, allocate time proportionally: 1 mark = ~1.5 minutes. If stuck on a 5-mark question beyond 8 minutes, move on and return later. Time management separates 85%+ scorers from the rest in Class 10 Mathematics.
  • Solve NCERT Exercises 6.1, 6.2, 6.3 fully; attempt Exercise 6.4 for Olympiad-level challenge.
  • Download CBSE sample papers and marking schemes from cbse.nic.in every year.
  • Practice at least 5 previous years' board questions on Triangles to spot recurring patterns.
  • Memorise exact theorem statements from NCERT; boards award marks for correct terminology.
  • In proofs, always write 'Given,' 'To Prove,' 'Construction,' 'Proof,' 'Hence proved' as separate headings.
  • Draw diagrams in pencil first, then darken with pen after verification; label all points clearly.
  • Revise formulas: BPT, similarity criteria, area ratio = (side ratio)², one day before the exam.

Frequently asked questions

What is the difficulty level and time limit for this CBSE Class 10 Mathematics Chapter 6 Triangles worksheet?+
Difficulty level is Moderate to Challenging, matching CBSE board exam standards. The suggested time limit is 90 minutes. This mirrors the actual exam duration per section and helps you build speed and accuracy under timed conditions.
How many marks is Chapter 6 Triangles worth in the CBSE Class 10 board exam?+
Chapter 6 Triangles typically accounts for 12–15 marks directly, including MCQs, short-answer, and one long-answer question. Additional indirect marks appear in construction and coordinate geometry questions that use similarity concepts, bringing the effective weightage to around 18 marks.
Which similarity criterion should I use if two angles of one triangle equal two angles of another triangle?+
Use the AAA (Angle-Angle-Angle) similarity criterion. When two angles are equal, the third angle is automatically equal because the sum of angles in a triangle is 180°. Therefore, all three corresponding angles are equal, proving the triangles are similar by AAA.
What is the converse of the Basic Proportionality Theorem, and how often does it appear in exams?+
The converse states: If a line divides two sides of a triangle proportionally, then the line is parallel to the third side. This converse appears frequently in both MCQs and short-answer questions, especially in proof-based problems. NCERT Exercise 6.2 covers it in detail.
How is the area ratio of two similar triangles related to their side ratio?+
The ratio of the areas of two similar triangles equals the square of the ratio of their corresponding sides. For example, if the side ratio is 3:4, the area ratio is 3²:4² = 9:16. This result extends to other linear measures like medians, altitudes, and perimeters.
Are all equilateral triangles similar, and why?+
Yes, all equilateral triangles are similar because each has all three angles equal to 60°. Since corresponding angles are equal, the AAA similarity criterion is satisfied. Their corresponding sides are proportional by definition of equilateral, confirming similarity.
What are the most common mistakes students make in Triangles chapter questions?+
Common mistakes include confusing corresponding sides, forgetting to square the ratio when finding areas, misapplying BPT (using it when lines are not parallel), omitting diagram labels, and not stating the similarity criterion before starting proofs. Careful reading and structured working prevent these errors.
Should I memorise all theorem proofs from NCERT Chapter 6 for the board exam?+
Yes, memorise proofs of the Basic Proportionality Theorem, its converse, the three similarity criteria, and the area theorem. CBSE often asks 'State and prove' questions worth 5 marks. Even if you cannot recall every step, writing the structure (Given, To Prove, diagram, key steps) earns significant partial credit.
How does CBSETUTOR.ai help with Chapter 6 Triangles doubts at ₹999 per month?+
CBSETUTOR.ai provides a 24×7 AI tutor that solves your Triangles doubts instantly via photo upload. Whether you are stuck on a BPT numerical or a similarity proof, upload your working and receive step-by-step feedback. The flat ₹999/month covers all classes (6–12) and subjects, with a 3-day free trial to start.
What is the best strategy to score full marks in case-study questions on Triangles?+
Read the passage carefully and underline key data like measurements, parallel lines, and angle values. Draw or label the given diagram with this data. Answer each sub-question independently; even if one is tough, attempt the rest. Each sub-question is usually 1 mark, so manage your 8-minute time budget across all four parts.

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