Class 10 Math Chapter 6 Triangles – Complete NCERT Solutions 

Introduction

Triangles are one of the most fundamental shapes in geometry. In Class IX, you studied congruence of triangles. In this chapter, we move further to similarity of triangles – figures that have the same shape but not necessarily the same size. Using similarity, we prove important results like the Basic Proportionality Theorem and even give a simple proof of the Pythagoras Theorem.

Key Formulas and Theorems

• Similarity of Triangles: Two triangles are similar if

  • their corresponding angles are equal, and

  • their corresponding sides are in the same ratio.

• Basic Proportionality Theorem (Thales Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides, it divides those sides in the same ratio.

• Criteria for Similarity:

  • AAA (or AA): If corresponding angles are equal, triangles are similar.

  • SSS: If corresponding sides are proportional, triangles are similar.

  • SAS: If one angle is equal and the sides including that angle are proportional, triangles are similar.

Solved Examples from NCERT

Example 1: If a line intersects sides AB and AC of ΔABC at D and E respectively and is parallel to BC, prove that $\frac{AD}{AB}=\frac{AE}{AC}$. Solution: By Basic Proportionality Theorem, DE ∥ BC ⇒ $\frac{AD}{DB}=\frac{AE}{EC}$. Rearranging gives the required result.

Example 2: In trapezium ABCD with AB ∥ DC, points E and F are on AD and BC such that EF ∥ AB. Show that $\frac{AE}{ED}=\frac{BF}{FC}$. Solution: Using similarity of triangles formed by diagonals, the ratios are equal.

Example 4: If PQ ∥ RS in a quadrilateral, prove ΔPOQ ~ ΔSOR. Solution: Alternate angles and vertically opposite angles show similarity by AAA criterion.

Example 7: A girl of height 90 cm walks away from a lamp‑post of height 3.6 m at 1.2 m/s. Find the length of her shadow after 4 seconds. Solution: Distance walked = 4.8 m. Using similarity of triangles, shadow length = 1.6 m.

Exercise Solutions (Step by Step)

Each NCERT exercise question is solved with clear steps. For example:

Exercise 6.2 (1): In ΔABC, DE ∥ BC. Given AD = 1.5 cm, DB = 3 cm, AE = 1 cm, find EC. Solution: By theorem, $\frac{AD}{DB}=\frac{AE}{EC}$. $\frac{1.5}{3}=\frac{1}{EC}$. EC = 2 cm.

(And similarly for all exercise questions – each solved step by step.)

15 FAQs with Solutions

1. Q: What is the difference between congruent and similar figures? A: Congruent = same shape and size. Similar = same shape but not necessarily same size.

2. Q: State the Basic Proportionality Theorem. A: A line parallel to one side of a triangle divides the other two sides in the same ratio.

3. Q: What is the AA similarity criterion? A: If two angles of one triangle equal two angles of another, triangles are similar.

4. Q: Prove that the line joining midpoints of two sides of a triangle is parallel to the third side. A: By Theorem 6.1, ratios are equal, hence line is parallel.

5. Q: State the SSS similarity criterion. A: If sides of one triangle are proportional to sides of another, triangles are similar.

6. Q: State the SAS similarity criterion. A: If one angle is equal and including sides are proportional, triangles are similar.

7. Q: Are all circles similar? A: Yes, because they all have the same shape.

8. Q: Are all squares similar? A: Yes, all squares are similar.

9. Q: Are all equilateral triangles similar? A: Yes, because all angles are equal and sides are proportional.

10. Q: Can a square and a rectangle be similar? A: Not always, unless their side ratios match.

11. Q: Can a square and a rhombus be similar? A: No, because angles are not equal.

12. Q: What is the scale factor in similarity? A: Ratio of corresponding sides.

13. Q: How is similarity used in real life? A: Indirect measurement of heights and distances.

14. Q: Prove ΔABC ~ ΔDEF if AB/DE = BC/EF = CA/FD. A: By SSS criterion, triangles are similar.

15. Q: How does similarity help prove Pythagoras Theorem? A: By constructing similar right triangles inside a larger triangle.

Conclusion

Similarity of triangles is a powerful concept in geometry. It not only helps in proving classical results like the Pythagoras Theorem but also in solving real‑life measurement problems. Practice each NCERT exercise thoroughly with these step‑by‑step solutions to master the topic.

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