Chapter-6

Triangles

 

Congruence of Triangles

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 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.
Can you think of some more examples of congruent figures?

Now, which of the following figures are not congruent to the square in

The large squares (ii) and (iii) are obviously not congruent to the one in
(i), but the square (iv) is congruent to the one given.
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

Cut out each of these triangles from Fig. (ii) to (v) and turn them around and try to cover Δ ABC. Observe that triangles in Fig. 7.4 (ii), (iii) and (iv) are congruent to Δ ABC while Δ TSU of Fig 7.4 (v) is not congruent to Δ ABC.
If Δ PQR is congruent to Δ ABC, we write Δ PQR
Δ ABC.
Notice that when Δ PQR
Δ ABC, then sides of Δ PQR fall on corresponding equal sides of Δ 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, Δ PQR
Δ ABC; but it will not be correct to write ΔQRP Δ ABC.
 Similarly, for
                      FD ↔ AB, DE ↔ BC  and EF ↔ CA
                      and  F ↔ A, D ↔ B  and E ↔ C
So, Δ FDE
Δ ABC but writing Δ DEF Δ ABC is not correct.
Give the correspondence between the triangle in Fig. 7.4 (iv) and Δ 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 .

Now, draw two triangles with one side 4 cm and one angle 50°. Are they congruent?

See that these two triangles are not congruent.
Repeat this activity with some more pairs of triangles.
So, equality of one pair of sides or one pair of sides and one pair of angles is not sufficient to give us congruent triangles.
What would happen if the other pair of arms (sides) of the equal angles are also equal?
BC = QR, B = Q and also, AB = PQ. Now, what can you say
about congruence of Δ ABC and Δ PQR?
Recall from your earlier classes that, in this case, the two triangles are congruent.
Verify this for Δ ABC and Δ PQR
Repeat this activity with other pairs of triangles. Do you observe that the equality of two sides and the included angle is enough for the congruence of triangles? Yes, it is enough.

This is the first criterion for congruence of triangles.
Axiom 7.1 (SAS congruence rule) : Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the  other triangle.
This result cannot be proved with the help of previously known results and so it is accepted true as an axiom (see Appendix 1).
Let us now take some examples.