Resource 2: Examples of entries of a student’s mathematical dictionary

Word/conceptCongruence in triangles
Where?Chapter 7, Class IX
Any meaning in everyday language?According to the dictionary, ‘congruent with’ means ‘suitable, agreeing’, but I have never used that or heard that (The Oxford Dictionary, 1997).
Mathematical explanation from textbook/teacher‘Equal in all respects’ or ‘figures [triangles in this case] whose shapes and sizes are both the same’ (NCRT textbook, Class IX, p. 109).
My explanationCongruent triangles are triangles that are identical in shape and size. Sometimes I have to turn them around or flip them over to actually see this. So the lengths of their sides and the angles will be the same for the congruent triangles. But not only that! Those same length sides and angles have to be in the same place in the triangle – that is what they call ‘corresponding’. When you cut out the triangles, you can place them on top of each other and they will be perfect copies. Congruent triangles are at the same time similar triangles, but similar triangles are not always congruent! Also, have a look at my entry for ‘similar triangles’ to see how these concepts are connected.
IllustrationThese triangles are congruent:
Word/conceptSimilarity of triangles
Where?Chapter 6, Class X
Any meaning in everyday language?Something that is like something else. What makes something like something else is often not made clear.
Mathematical explanation from textbook/teacher

‘Two triangles are similar, if

  • i.their corresponding angles are equal and
  • ii.their corresponding sides or in the same ratio (or proportion).’
(NCRT textbook, Class X, p. 123)
My explanation

Note to myself: do not mix up with congruent!

Similar triangles have the same shape but not necessarily the same size. If they do have the same shape and the same size they are similar as well as congruent. So they have to be similar in shape, but not identical in size. What makes them similar is that the shapes are a proportional enlargement or reduction of each other.

That means that the proportion or ratio of all the corresponding sides will be the same (for example, the second triangle has sides twice the size of the first). And that is why these criteria about similarity of triangles make sense.

IllustrationThese triangles are similar:

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