Chapters

Triangles

NCERT Class 10 Mathematics — Chapter 6

In mathematics the art of asking questions is more valuable than solving problems. — Georg Cantor

6.1 Introduction

This chapter studies similar figures and similar triangles, and uses similarity to prove results about sides and areas.

6.2 Similar Figures

Two figures are similar if they have the same shape (not necessarily the same size). Corresponding angles are equal and corresponding sides are in the same ratio.

For polygons, equal angles + proportional sides ⇒\Rightarrow similar.

6.3 Similarity of Triangles

Two triangles are similar if their corresponding angles are equal (AAA), or if two angles of one equal two angles of the other.

If △ABC∼△DEF\triangle ABC \sim \triangle DEF, then

ABDE=BCEF=CAFD\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}

6.4 Criteria for Similarity

  • AAA (or AA): all corresponding angles equal
  • SSS: sides in proportion
  • SAS: one angle equal and including sides proportional

Basic Proportionality Theorem (Thales): if a line parallel to one side of a triangle intersects the other two sides, it divides those sides in the same ratio.

Converse: if a line divides two sides in the same ratio, it is parallel to the third side.

Chapter Summary

  • Similar: same shape, sides in proportion
  • AAA, SSS, SAS criteria
  • BPT and its converse

Exercises (NCERT)

  • Exercise 6.1 — similar figures
  • Exercise 6.2 — similarity criteria, BPT
  • Exercise 6.3 — applications
  • Miscellaneous Exercise