6. Triangles

7 topics2 h

Chapter 6 of NCERT Mathematics for Class 10: Triangles. Three sections — Same shape, different size, The Basic Proportionality Theorem and Criteria for similarity. Seven videos, 2 h in all.

Worked answers to the 29 exercise questions in this chapter: Exercise 6.1 · Exercise 6.2 · Exercise 6.3

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Part 1

Same shape, different size

12 minWhat similarity asks for that congruence does notSimilarity is not a second relation standing beside congruence. It is the same physical test - lay one figure on the other - with a single demand withdrawn, and the whole of what follows is a consequence of which demand went.12 min · 13 parts14 minThe two conditions polygons must both meetTwo figures are similar when their matching corners are equal AND their matching sides are in one ratio. The word doing the work is and: a square against a rectangle passes the first clause on its own, a square against a rhombus passes the second on its own, and neither pair is similar.14 min · 14 parts

Part 2

The Basic Proportionality Theorem

16 minA line drawn parallel to a side cuts the other two in matching ratiosA line drawn across a triangle parallel to one of its sides cuts the other two in the same ratio. That is not something you discover by measuring carefully - it is squeezed out of a single fact about area, and the parallel hypothesis is spent in exactly one place.16 min · 14 parts16 minTurning it around: matching ratios force the line to be parallelA statement and its converse are two different questions, and this one is settled by a fact about a segment: a ratio names exactly one point of it. Everything else in the proof is scaffolding around that.16 min · 14 parts

Part 3

Criteria for similarity

16 minAAA and AA: equal angles are enough on their ownSame shape means matching angles equal AND matching sides in one ratio - two independent demands, as a rectangle beside a square proves. For triangles the independence collapses: the angles alone force the ratio, and one construction shows why.16 min · 14 parts15 minSSS: sides in proportion drag the angles into agreementA square and a rhombus have all four sides in one ratio and not one corner in agreement. Do the same thing to triangles and the corners have nowhere to go - and the proof is the previous construction, walked through the door in the other direction.15 min · 13 parts15 minSAS: one angle plus the two sides that enclose itTwo triangles, both side ratios exactly a half, both marked angles exactly equal - and they are not the same shape. Everything turns on which vertex the mark is sitting at.15 min · 13 parts

What you will be able to do

  • State the difference between two figures being congruent and being similar, in terms of which of shape and size is being held fixed
  • State the two requirements a pair of polygons must both meet to be called similar, and say which one is about angles and which about lengths
  • Perform the construction of Activity 2 and report the two ratios it produces
  • State the converse of the Basic Proportionality Theorem and distinguish its hypothesis from the forward theorem's
  • Write a similarity statement for two triangles with the vertices in the correct order, and say which reorderings are still valid
  • State the question SSS answers, and say why the answer is not obvious given the quadrilateral counter-example
  • State the SAS criterion for similarity, with the enclosure condition stated explicitly

What this chapter assumes you already know

  • Congruence of triangles from Class IX, and the fact that congruent figures can be laid one on the other and coincide
  • That a circle is fixed by its radius, a square by its side, an equilateral triangle by its side
  • Reading a ratio of two lengths as a single number, and simplifying it
  • Measuring an angle with a protractor, and knowing that an angle is unchanged when a figure is scaled
  • Naming corresponding vertices of two polygons and keeping the pairing straight
  • Writing a ratio of two lengths and testing whether several ratios are equal
  • The angle properties of a square, a rectangle and a rhombus from earlier classes
  • That the interior angles of any quadrilateral total 360°

It builds directly on A fixed step between neighbours is the whole definition.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Similar means roughly alike."
  • "Congruent and similar are two alternatives to choose between."
  • "If it looks similar, it is."
  • "All triangles are similar, since they are all triangles."
  • "Bigger means not similar."
  • "Enlarging a photograph opens out the angles."
  • "Similar figures must be the same way up."
  • "Equal angles is the real test; sides follow."