Triangles (Congruence) Class 9

📝Triangles (Congruence)

Triangle

➤ A closed figure formed by three intersecting lines is called a triangle. (‘Tri’ means ‘three’). A triangle has three sides, three angles, and three vertices.

Triangle ABC, denoted as

$triangle ABC$ (see Fig.);

$AB, BC, CA$ are the three sides,

$angle A, angle B, angle C$ are the three angles, and

$A, B, C$ are three vertices.

Congruent Figures

➤ Two figures are congruent if they are of the same shape and of the same size.

● Two circles of the same radii are congruent.

● Two squares of the same sides are congruent.

Congruence of Triangles

$trianglePQR$ is congruent to

$triangleABC$ , we write

$trianglePQR ≅ triangleABC$ .

Notice that when

$trianglePQR≅ triangleABC$ , then sides of

$trianglePQR$ fall on corresponding equal sides of

$triangleABC$ and so is the case for the angles.

That is, PQ covers AB, QR covers BC, and RP covers CA;

$angleP$ covers

$angleA$ ,

$angleQ$ covers

$angleB$ and

$angleR$ covers

$angleC$ . 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,

$trianglePQR ≅ triangleABC$ ; but it will not be correct to write

$triangleQRP ≅ triangleABC$ .

Similarly, for Fig. (iii),

$↔$ AB, DE

$↔$ BC and EF

$↔$ CA and F

$↔$ A, D

$↔$ B and E

$↔$ C

So,

$triangleFDE ≅ triangle ABC$ but writing

$triangleDEF ≅ triangleABC$ is not correct.

Note that in congruent triangles corresponding parts are equal and we write in short ‘CPCT’ for corresponding parts of congruent triangles.

Criteria for Congruence 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 (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.

$↔$

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