You must have observed that two copies of your photographs of the same size are identical. Similarly, two bangles of the same size, two ATM cards issued by the same bank are identical.
You may recall that on placing a one rupee coin on another minted in the same year, they cover each other completely.
Do you remember what such figures are called? Indeed they are called congruent figures (‘congruent’ means equal in all respects or figures whose shapes and sizes are both the same).
Now, draw two circles of the same radius and place one on the other. What do you observe? They cover each other completely and we call them as congruent circles.
Repeat this activity by placing one square on the other with sides of the same measure (see Fig. 7.2) or by placing two equilateral triangles of equal sides on each other. You will observe that the squares are congruent to each other and so are the equilateral triangles.
You may wonder why we are studying congruence. You all must have seen the ice tray in your refrigerator. Observe that the moulds for making ice are all congruent.
The cast used for moulding in the tray also has congruent depressions (may be all are rectangular or all circular or all triangular). So, whenever identical objects have to be produced, the concept of congruence is used in making the cast.
Sometimes, you may find it difficult to replace the refill in your pen by a new one and this is so when the new refill is not of the same size as the one you want to remove. Obviously, if the two refills are identical or congruent, the new refill fits
So, you can find numerous examples where congruence of objects is applied in daily life situations.
Now, which of the following figures are not congruent to the square in Fig `7.3` (i) :
The large squares in Fig. 7.3 (ii) and (iii) are obviously not congruent to the one in Fig 7.3 (i), but the square in Fig 7.3 (iv) is congruent to the one given in Fig 7.3 (i).
Let us now discuss the congruence of two triangles.
You already know that two triangles are congruent if the sides and angles of one triangle are equal to the corresponding sides and angles of the other triangle.
Now, which of the triangles given below are congruent to triangle ABC in Fig. 7.4 (i)?
Cut out each of these triangles from Fig. `7.4` (ii) to (v) and turn them around and try to cover `Delta ABC`. Observe that triangles in Fig. 7.4 (ii), (iii) and (iv) are congruent to `Delta ABC` while `Delta TSU` of Fig `7.4` (v) is not congruent to `Delta ABC`.
If `Delta PQR` is congruent to `Delta ABC`, we write `Delta PQR ≅ Delta ABC`. Notice that when `Delta PQR ≅ Delta ABC`, then sides of `Delta PQR` fall on corresponding equal sides of `Delta ABC` and so is the case for the angles.
That is, `PQ` covers `AB, QR` covers `BC` and `RP` covers `CA; ∠ P` covers `∠ A,∠ Q` covers `∠ B` and `∠ R` covers `∠ C`. Also, there is a one-one correspondence between the vertices. That is, P corresponds to `A, Q` to `B, R` to `C` and so on which is written as
`P ↔ A, Q ↔ B, R ↔ C`
Note that under this correspondence, `Delta PQR ≅ Delta ABC`; but it will not be correct to write `Delta QRP ≅ Delta ABC`.
Similarly, for Fig. 7.4 (iii),
`FD ↔ AB, DE ↔ BC` and `EF ↔ CA`
and `F ↔ A, D ↔ B` and `E ↔ C`
So, `Delta FDE ≅ Delta ABC` but writing `Delta DEF ≅ Delta ABC` is not correct.
Give the correspondence between the triangle in Fig. 7.4 (iv) and `Delta ABC`.
So, it is necessary to write the correspondence of vertices correctly for writing of congruence of triangles in symbolic form.
Note that in congruent triangles corresponding parts are equal and we write in short ‘CPCT’ for corresponding parts of congruent triangles.