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**Congruence of Triangles**

**7.1 INTRODUCTION**

You are now ready to learn a very important geometrical idea, Congruence. In particular, you will study a lot about congruence of triangles. To understand what congruence is, we turn to some activities. The relation of two objects being congruent is called congruence. For the present, we will deal with plane figures only, although congruence is a general idea applicable to three-dimensional shapes also. We will try to learn a precise meaning of the congruence of plane figures already known.

**7.2 CONGRUENCE OF PLANE FIGURES**

Look at the two figures given here (Fig 7.3). Are they congruent? You can use the method of superposition. Take a trace-copy of one of them and place it over the other. If the figures cover each other completely, they are congruent. Alternatively, you may cut out one of them and place it over the other. Beware! You are not allowed to bend, twist or stretch the figure that is cut out (or traced out). In Fig 7.3, if figure F1 is congruent to figure F2 , we write F1 ≅ F2.

**7.3 CONGRUENCE AMONG LINE SEGMENTS**

When are two line segments congruent? Observe the two pairs of line segments given here (Fig 7.4). Use the ‘trace-copy’ superposition method for the pair of line segments in [Fig 7.4(i)]. Copy CDand place it on AB. You find that CD covers AB, with C on A and D on B. Hence, the line segments are congruent. We write AB ≅ CD. Repeat this activity for the pair of line segments in [Fig 7.4(ii)]. What do you find? They are not congruent. How do you know it? It is because the line segments do not coincide when placed one over other.

You should have by now noticed that the pair of line segments in [Fig 7.4(i)] matched with each other because they had same length; and this was not the case in [Fig 7.4(ii)]. In view of the above fact, when two line segments are congruent, we sometimes just say that the line segments are equal; and we also write AB = CD. (What we actually mean is AB≅ CD)

**7.4 CONGRUENCE OF TRIANGLES**

We saw that two line segments are congruent where one of them, is just a copy of the other. Similarly, two angles are congruent if one of them is a copy of the other. We extend this idea to triangles.

Two triangles are congruent if they are copies of each other and when superposed, they cover each other exactly.

ΔABC and ΔPQR have the same size and shape. They are congruent. So, we would express this as

ΔABC ≅ ΔPQR

This means that, when you place ΔPQR on ΔABC, P falls on A, Q falls on B and R falls on C, also falls along AB , QR falls along BCand PR falls along AC. If, under a given correspondence, two triangles are congruent, then their corresponding parts (i.e., angles and sides) that match one another are equal. Thus, in these two congruent triangles, we have:

Corresponding vertices : A and P, B and Q, C and R.

Corresponding sides : ABand PQ, BC and QR , AC and PR .

Corresponding angles : ∠A and ∠P, ∠B and ∠Q, ∠C and ∠R.

If you place ΔPQR on ΔABC such that P falls on B, then, should the other vertices also correspond suitably? It need not happen! Take trace, copies of the triangles and try to find out.

This shows that while talking about congruence of triangles, not only the measures of angles and lengths of sides matter, but also the matching of vertices. In the above case, the correspondence is

A ↔P, B ↔ Q, C ↔ R

We may write this as ABC ↔PQR

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