Facts That Matter
If two geometrical figures have exactly the same shape and size then they are called congruent figures.
Note: (i) Two line segments are congruent only when their lengths are equal.
(ii) Two angles are congruent only when their degree measures are equal. ∠ABC ≌ ∠PQR
(iii) The symbol ‘≌’ is used to represent congruence.
Two triangles are congruent if and only if one of them can be made to superimpose on the other, such as to cover it exactly.
In the figure, ΔABC ≌ ΔDEF.
Note: (i) When ΔABC ≌ ΔDEF, then sides of ΔDEF fall on corresponding equal sides of ΔABC, i.e.
DE covers AB or DE ↔ AB; EF covers BC or EF ↔ BC and FD covers CA or FD ↔ CA.
(ii) In case ΔABC ≌ ΔDEF,
∠D covers ∠A or ∠D ↔ ∠A.
∠E covers ∠B or ∠E ↔ ∠B.
∠F covers ∠C or ∠F ↔ ∠C.
(iii) In case ΔABC ≌ ΔDEF
D corresponds to A or D ↔ A.
E corresponds to B or E ↔ B.
F corresponds to C or F ↔ C.
(iv) In congruent triangles, corresponding parts are equal and we write in short ‘c.p.c.t.’ “for Corresponding Parts of Congruent Triangles”.
CRITERIA FOR CONGRUENCE OF TRIANGLES
1. SAS Criteria:
If two triangles are such that two sides and the included angle of the one equal to the corresponding sides and the included angle of the other, then the triangles are congruent.
In the figure, we have
AB = DE, BC = DF and ∠B = ∠E
∴ ΔABC ≌ ΔDEF.
Note: It is necessary to write the correspondence of vertices correctly for writing of congruence of triangles in symbolic form.
In case of ΔABC ≌ ΔDEF, we have
A ↔ D, B ↔ E and C ↔ F.
II . ASA Criteria:
If two triangles are such that two angles and the included side of one are equal to the corresponding two angles and the included side of the other, then the two triangles are congruent. In the figure, we have
∠A = ∠D; ∠B = ∠E and AB = DE
∴ ΔABC ≌ ΔDEF.
Note: In case, in two triangles, two pairs of angles and one pair of corresponding sides are equal (even if the side is not included between the corresponding equal pairs of angles), the triangles are congruent.
This is because the sum of three angles of a triangle is 180°, therefore, if two pairs of angles are equal, the third pair is also equal (180° - sum of equal angles). Thus, two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal.
We may call it as the AAS congruence rule.