Textbook chapter 6
Use proportional sides and equal angles to recognise similar triangles. Apply a parallel-line theorem and justify your geometric reasoning.
Textbook exercise solutions
- Exercise 6.1 — 3 question groups
- Exercise 6.2 — 10 question groups
- Exercise 6.3 — 16 question groups
Videos, notes and tests
English · Session 2026–27 · NCERT-aligned study guide
Chapter overview
Use proportional sides and equal angles to recognise similar triangles. Apply a parallel-line theorem and justify your geometric reasoning.
Before you begin: Angles in triangles, parallel lines, areas of triangles and ratios.
Key concepts · Solved examples · Practice · Mind map · Timed chapter test
Learn the key concepts
Similarity and correspondence
Similar figures have the same shape. Similar triangles have equal corresponding angles and proportional corresponding sides. If ΔABC ∼ ΔPQR, the order tells you A↔P, B↔Q, C↔R. Write ratios in that order.
Basic Proportionality Theorem
If D lies on AB, E on AC and DE ∥ BC, then AD/DB=AE/EC. Proof: triangles ADE and BDE have the same altitude to AB, so their area ratio is AD/DB. Similarly area(ADE)/area(CDE)=AE/EC. Triangles BDE and CDE have common base DE and equal altitudes because BC ∥ DE; hence equal areas. Equating the two ratios proves the result. The converse detects parallel lines from equal divisions.
Similarity tests
AA: two matching angles suffice. SSS: all three corresponding side ratios agree. SAS: the two side ratios agree and their included angles agree. An arbitrary equal angle with two proportional sides is not enough for SAS.
Formula and result checklist
- DE ∥ BC ⇒ AD/DB = AE/EC and AD/AB = AE/AC.
- For ΔABC ∼ ΔPQR: AB/PQ = BC/QR = AC/PR.
Solved examples
Example 1
In ΔABC, DE ∥ BC, AD=4, DB=6 and AE=5. Find EC.
By BPT, 4/6=5/EC. Thus 4EC=30 and EC=7.5 units.
Example 2
Triangles have corresponding sides (3,4,5) and (6,8,10). Are they similar?
3/6=4/8=5/10=1/2. All corresponding sides are proportional, so the triangles are similar by SSS.
Example 3
ΔABC ∼ ΔPQR, AB=5, PQ=8 and BC=7. Find QR.
AB/PQ=BC/QR gives 5/8=7/QR. Therefore QR=56/5=11.2 units.
Try it yourself
Work on paper before opening each answer. Show the reasoning and include units when needed.
1. DE ∥ BC, AD=3, DB=2 and AE=6. Find EC.
Check answer 1
3/2=6/EC, so EC=4.
2. Two triangles each have angles 50° and 60°. Are they similar?
Check answer 2
Yes, by AA; the third angle is 70° in both.
3. D,E lie on AB,AC with AD/DB=2/3 and AE/EC=4/6. What follows?
Check answer 3
DE ∥ BC by the converse of BPT.
Common mistakes to avoid
- Do not mix a part-to-part ratio with a part-to-whole ratio.
- Similarity is not the same as congruence; the scale factor need not be 1.
Chapter mind map
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Take the 6-question, 10-minute chapter practice test. Each correct answer earns 1 mark; there is no negative marking. See explanations after submission.
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Read the corresponding chapter in the official NCERT textbook, then complete its exercises on paper. These NSL explanations and original practice questions support your textbook study; they are not an exercise-by-exercise solution manual.
Official CBSE syllabus 2026–27 · All chapters · Previous chapter · Next chapter
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