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Congruence of Triangles

Congruent means coinciding exactly when superimposed. Two geometrical figures are said to be congruent if they have exactly the same shape and size and the property is called congruence.
 

Congruence of Two Line Segmentsb :
Two line-segments will be congruent if and only if their lengths are equal.
 

Congruence of Two Line Segments


Two line-segments PQ and AB are congruent if AB = PQ

Two triangles are congruent, if all the sides and all the angles of one are equal to the corresponding sides and angles of the other.
 

Congruence of Two Angles

When ΔABC is congruent to ΔPQR we get six equalities; three of the corresponding sides and three of the corresponding angles. Suppose ΔABC superimposes ΔPQR exactly and the vertices of ΔABC fall on the vertices of ΔPQR in the order A P, B Q, C R or we may write in short as ABC PQR

Then we have the six equalities:
(i) AB = PQ
(ii) BC = QR
(iii) CA = RP
(iv)
ABC = PQR or B = Q
(v)
BCA = QRP or C = R
(vi)
CAB = RPQ or A = P

In the two triangles ABC and PQR, there will be six matching or correspondence between their vertices:

A
P B Q C R or ABC PQR
A
Q B R C P or ABC QRP
A
R B P C Q or ABC RPQ
A
P B R C Q or ABC PRQ
A
Q B P C R or ABC QPR
A
R B Q C P or ABC RQP

If ΔABC is congruent to ΔPQR, then one of the matching ones will superimpose exactly on the other. The corresponding side angles of the two triangles will be equal (congruent).

Two triangles are congruent if and only if there exists a correspondence between their vertices such that the corresponding sides and angles of the two triangles are equal.

 

 
Note : If ΔABC is congruent to ΔPQR and the correspondence ABC PQR makes the six pairs of corresponding parts of the two triangles congruent, then we write ΔABC ΔPQR. If two triangles are congruent. Their corresponding parts (angles and sides) will be equal and can be written as c.p.c.t. i.e. corresponding parts of congruent triangles.




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