NCERT Solutions for Class 10 Maths Chapter 6 Triangles

Looking for NCERT Solutions for Class 10 Maths Chapter 6 (Triangles)? This page tells you which exercise holds your question and what method it wants, before you open a solution page.

Chapter 6 studies figures that have the same shape but not necessarily the same size — similar figures. It builds two skills: dividing the sides of a triangle by a parallel line (the Basic Proportionality Theorem) and proving two triangles similar with a criterion. Similarity is also how heights of mountains and distances of far objects are measured indirectly, without a tape.

Below is a directory of the chapter’s three exercises — Exercise 6.1, 6.2 and 6.3 — with what each tests, followed by the methods the chapter runs on. The solved exercise pages sit one level down.

Which exercise holds your question: the directory

All 29 questions of Chapter 6 sit in three exercises. The table below names each exercise, how many questions it holds, what it tests and the method it wants; the solved pages link from its rows.

• Exercise 6.1 (3 questions) tests the definition of similar figures: fill in blanks on which shapes are similar, give your own similar and non-similar examples, and judge from a figure whether two quadrilaterals match. The method is the definition itself — equal corresponding angles and proportional sides.
• Exercise 6.2 (10 questions) tests the Basic Proportionality Theorem (Theorem 6.1) and its converse (Theorem 6.2). The first two questions are the chapter’s numerical BPT work — find an unknown side, or test whether a segment is parallel by comparing ratios. Questions 3–10 prove parallel-line chains, recover the midpoint theorems, and prove the diagonals property of a trapezium.
• Exercise 6.3 (16 questions) tests choosing the right similarity criterion. Question 1 asks you to name the criterion for each pair of triangles in a figure; the rest chase equal angles, or work through altitudes, angle bisectors and medians, and close with the pole-and-tower shadow word problem (Question 15).

Basic Proportionality Theorem and its converse: the first tool

Theorem 6.1 (Thales’ theorem) says a line drawn parallel to one side of a triangle divides the other two sides in the same ratio. With \(DE \parallel BC\) in \(\triangle ABC\), that ratio is \( \frac{AD}{DB} = \frac{AE}{EC} \).

Theorem 6.2 is its converse: if a line divides two sides of a triangle in the same ratio, that line is parallel to the third side. Two ratio forms are worth knowing — the segment form \( \frac{AD}{DB} = \frac{AE}{EC} \) and the whole-side form \( \frac{AD}{AB} = \frac{AE}{AC} \) shown in the chapter’s Example 1.

Exercise 6.2 runs entirely on these two theorems. Questions 1 and 2 are its only numerical questions; Questions 3–10 are proofs routed through either one.

AAA, AA, SSS and SAS: picking the right similarity criterion

Exercise 6.3 checks your eye for which criterion fits the triangles you are given. One condition is all each needs.

Two rules protect marks here. The similarity statement must keep corresponding vertices in order: once \( \triangle ABC \sim \triangle DEF \) is true, writing \( \triangle ABC \sim \triangle EDF \) asserts the wrong angle correspondences (textbook p.86). And for triangles only one condition — angles OR sides — is enough, because one implies the other.

Polygons need both: a square and a rectangle have equal angles but not proportional sides, so they are not similar.

Mistakes that cost marks in triangle similarity

• Vertices written out of order. \( \triangle ABC \sim \triangle EDF \) silently claims the wrong angles correspond. Match vertices the way the figure demands (textbook p.86).
• Comparing the ratio the wrong way round in Exercise 6.2 Q2. To test whether a segment is parallel, both ratios must be set the same way and both sides checked; one reversed fraction gives a false verdict.
• Checking only angles OR only sides for polygons. A square and a rhombus share proportional sides but different angles; a square and a rectangle share angles but not sides. Polygons need both, so neither pair is similar (textbook p.75–78).
• Using SAS when the equal angle is not the included angle. SAS needs the equal angle between the two proportional sides; otherwise the criterion does not apply (textbook p.90).
• Confusing similarity with congruence. Congruent figures are always similar, but similar figures need not be congruent — same shape, any size (textbook p.75).
• Mixing units in Exercise 6.3 Q15. The girl’s height is given in cm and must be converted to metres before the shadow ratio is set up.

How to route your homework to the right exercise

• A conceptual question — “are these figures similar?”, blanks, give examples — goes to Exercise 6.1.
• A question that mentions a line parallel to a side of a triangle, or asks whether two segments inside a triangle are parallel, goes to Exercise 6.2.
• A proof that two triangles are similar, or a request to name the criterion, goes to Exercise 6.3 — where Question 15’s pole-and-tower shadow problem lives.

The solved pages one level down carry the full working; no question is answered on this page. If you want to verify any definition or theorem, open the official NCERT Class 10 Mathematics textbook (Chapter 6).

Then move through the rest of the syllabus — the neighbouring chapters Arithmetic Progressions and Coordinate Geometry, the Class 10 Maths hub, the Class 10 notes folder, and the wider CBSE notes library.

Triangles chapter FAQs

Which exercise has the pole and tower shadow question?

Exercise 6.3, Question 15. It is the chapter’s indirect measurement word problem, solved by proving two right triangles similar inside one figure.

Which exercise covers the Basic Proportionality Theorem (Thales theorem)?

Exercise 6.2. It drills Theorem 6.1 (Thales’ theorem) and its converse, Theorem 6.2.

What is the difference between Exercise 6.1 and Exercise 6.2?

Exercise 6.1 tests the definition of similar figures — which shapes are similar and why. Exercise 6.2 moves to the parallel-line theorem that divides triangle sides in the same ratio.

How many questions are there in Exercise 6.3?

Sixteen. Most are proofs using the similarity criteria, including the shadow question and several altitude- and median-based problems.

Is RHS a similarity criterion?

Yes. The note to the reader (textbook p.98) states that two right triangles with hypotenuse and one side proportional are similar — the RHS similarity criterion.

Reference: NCERT textbooks (CBSE).