📐 CBSE · Class 10 · Mathematics · Chapter 6

Chapter 6:
Triangles

Complete chapter resources for CBSE Class 10 Maths — similarity criteria, Basic Proportionality Theorem, Pythagoras theorem proofs, sample questions, previous year board questions, and instant AI question paper generation.

5Topics
6–10Board marks
8Sample questions
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Key Theorems & Formulas — Chapter 6
  • BPT (Thales): DE ∥ BC ⟹ AD/DB = AE/EC
  • AA Similarity: ∠A = ∠P, ∠B = ∠Q ⟹ △ABC ~ △PQR
  • Area ratio: ar(△ABC)/ar(△PQR) = (AB/PQ)²
  • Pythagoras: AC² = AB² + BC² (∠B = 90°)
  • Converse Pythagoras: AC² = AB² + BC² ⟹ ∠B = 90°
Chapter Overview

What this chapter covers

Chapter 6 of NCERT Class 10 Mathematics is the most geometry-heavy chapter in the syllabus. It introduces the concept of similar triangles — triangles that have the same shape but not necessarily the same size. Two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional. The chapter builds this idea rigorously through theorems, proofs, and a variety of application problems.

The chapter opens with the Basic Proportionality Theorem (also called Thales' theorem): a line drawn parallel to one side of a triangle divides the other two sides in the same ratio. Its converse is equally important. The similarity criteria — AA, SAS, and SSS — then give students systematic tools to establish similarity between triangles. Once similarity is established, the ratio of areas equals the square of the ratio of corresponding sides, which is a key result for board word problems.

The chapter concludes with the Pythagoras theorem and its converse, both proved using triangle similarity. The Pythagoras theorem proof (using the altitude to the hypotenuse) is a perennial 4–5 mark board question. Students must be comfortable writing formal geometry proofs, identifying similarity ratios, and applying these results to real-world scenarios involving heights, distances, and areas.

Topics

What's inside Chapter 6

As per NCERT Class 10 Mathematics (CBSE syllabus)

Topic 1
Similar Figures & Similar Triangles
Definition of similarity — same shape, proportional sides, equal corresponding angles. Distinguishing congruence from similarity. Introduction to scale factor.
Topic 2
Basic Proportionality Theorem (BPT)
Thales' theorem: a line parallel to one side of a triangle divides the other two sides proportionally. Proof of BPT and its converse. Application to find unknown side lengths.
Topic 3
Criteria for Similarity of Triangles
AA (Angle-Angle), SAS (Side-Angle-Side), and SSS (Side-Side-Side) criteria. Writing formal similarity statements and identifying corresponding vertices correctly.
Topic 4
Areas of Similar Triangles
Theorem: the ratio of areas of two similar triangles equals the square of the ratio of their corresponding sides. Application problems involving area ratios and side ratios.
Topic 5
Pythagoras Theorem & Its Converse
Proof of the Pythagoras theorem using similarity of triangles. Converse: if a² + b² = c², the angle opposite c is 90°. Applications to right-triangle problems and distance calculations.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Class 9 Ch 7 · Triangles
Congruence criteria (SAS, ASA, SSS, RHS) and triangle properties
Class 9 Ch 9 · Quadrilaterals
Mid-point theorem, parallel lines, and ratio results
Chapter 6 Triangles
Leads to
Ch 8 · Introduction to Trigonometry
Trigonometric ratios defined on right triangles; Pythagoras is foundational
Ch 10 · Circles
Tangent-length theorem and chord proofs use triangle similarity
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 10 board papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Objective 1 1–2 Identify similarity criterion, find a side ratio, or state BPT
Very Short Answer 2 1 Find unknown side using BPT or similarity ratio; verify Pythagoras
Short Answer / Proof 3 1 Prove triangles similar and deduce a ratio; area of similar triangles
Long Answer / Proof 4–5 1 Proof of Pythagoras theorem; BPT proof; multi-step application
Total (approximate) 6–10 4–5 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 10 Maths Chapter 6. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
In △ABC, DE ∥ BC. If AD = 3 cm, DB = 6 cm, and AE = 2 cm, what is the length of EC? (a) 1 cm (b) 3 cm (c) 4 cm (d) 6 cm
Q2 Easy 2 marks Short Answer
In △ABC and △DEF, ∠A = ∠D = 50°, ∠B = ∠E = 70°. State the similarity criterion that proves △ABC ~ △DEF. Write the corresponding sides in ratio form.
Q3 Medium 2 marks Short Answer
In △PQR, ST ∥ QR. If PS = 4 cm, SQ = 8 cm, and PT = 3 cm, find TR. Also verify that ST divides PQ and PR in the same ratio.
Q4 Medium 3 marks Short Answer
The areas of two similar triangles △ABC and △PQR are 64 cm² and 100 cm² respectively. If AB = 8 cm, find the length of PQ. Also find the ratio of their perimeters.
Q5 Medium 3 marks Proof
In △ABC, ∠BAC = 90° and AD ⊥ BC where D lies on BC. Prove that: (i) △ABD ~ △CAD (ii) AB² = BD × BC
Q6 Hard 4 marks Proof
Prove the Pythagoras Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Q7 Hard 4 marks Word Problem
A vertical pole 6 m tall casts a shadow 4 m long on level ground. At the same time, a nearby building casts a shadow 28 m long. Using the concept of similar triangles, find the height of the building. Also state which similarity criterion you used.
Q8 Hard 5 marks Case-Based
Two triangles ABC and DEF are such that AB/DE = BC/EF = CA/FD = 2/3. (i) Are the triangles similar? Name the criterion. (ii) If the perimeter of △ABC is 30 cm, find the perimeter of △DEF. (iii) If the area of △ABC is 36 cm², find the area of △DEF. (iv) If BC = 8 cm, find EF.
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 10 Maths board papers — Triangles chapter.

Board 20223 marks
In the given figure, if △ABC ~ △DEF and their sides are in ratio 3 : 5, find the ratio of their areas. Also, if the area of △DEF is 150 cm², find the area of △ABC. (Delhi 2022)
Board 20235 marks
Prove that in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. (All India 2023)
Board 20202 marks
In △ABC, DE ∥ BC where D and E are points on AB and AC respectively. If AD/AB = 3/7, find the value of (area of △ADE) / (area of trapezium BCED). (CBSE 2020)
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FAQ

Questions teachers ask

How many marks does Triangles carry in the CBSE Class 10 board exam? +
Triangles typically carries 6–10 marks spread across 3–5 questions — one 1-mark MCQ on similarity criteria or BPT, one 2-mark short answer on ratio/length finding, one 3-mark proof or application, and one 4–5 mark long-answer proof or word problem. The Pythagoras theorem proof appears in nearly every board paper.
Which similarity criteria are most important for CBSE Class 10 board exams? +
All five criteria — AA, SAS, SSS, RHS, and ASA — are in the syllabus, but AA (Angle-Angle) and SAS (Side-Angle-Side) similarity appear most frequently in board questions. AA is particularly important because it only requires establishing two equal angles, making it faster to apply in proof-type questions.
Is the proof of the Pythagoras theorem required for board exams? +
Yes. The NCERT proof of the Pythagoras theorem — using the similarity of triangles formed by the altitude to the hypotenuse — is a high-frequency 4–5 mark board question. Students must know this proof by heart. The converse (if a² + b² = c², the angle opposite c is 90°) is also tested as a 2-mark short answer.
What is the Basic Proportionality Theorem and how is it tested? +
The Basic Proportionality Theorem (Thales' theorem) states: if a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. Board questions either ask for the proof (3–5 marks) or give numerical values and ask students to find an unknown side length using the ratio (1–2 marks). The converse is also tested.
How do I generate a custom question paper for Triangles using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 10 Mathematics, select Chapter 6 (Triangles), set your preferred question-type mix (MCQ, proof, application, word problem) and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.