Here we will learn different

criteria for congruency of triangles.

I. SAS (Side-Angle-Side) Criterion:

If two triangles have two sides of one equal to two sides of

the other, each to each, and the angles included by those sides are equal then

the triangles are congruent.

Here in ∆KLM and ∆XYZ,

KL = XY, LM = YZ and ∠L = ∠Y

Therefore, ∆KLM ≅ ∆XYZ.

**Note:** It is necessary for the included angles to be equal

for congruency. If in the above figure, ∠L ≠ ∠Y and ∠L = ∠X, the triangle may not

be congruent.

II. AAS (Angle-Angle-Side) Criterion:

If two triangles have two angles of one equal to two angles of

the other, each to each, and any side of the one equal to the corresponding

side of the other, then the triangles are congruent.

Here in ∆KLM and ∆XYZ,

∠L = ∠Y, ∠M = ∠Z and KM = XZ.

Therefore, ∆KLM ≅ ∆XYZ.

III. SSS (Side-Side-Side) Criterion:

If two triangles have three sides of one equal to three

sides of the other, the triangles are congruent.

Here in ∆KLM and ∆XYZ,

KL = XY, LM = YZ and KM = XZ.

Therefore, ∆KLM ≅ ∆XYZ.

IV: RHS (Right Angle-Hypotenuse-Side) Criterion:

If two right-angled triangles have their hypotenuses equal

and one side of one equal to one side of the other, the triangles are

congruent.

IV: RHS (Right Angle-Hypotenuse-Side) Criterion:

If two right-angled triangles have their hypotenuses equal

and one side of one equal to one side of the other, the triangles are

congruent.

Here, ∠L = ∠Y = 90°, KM = XZ and KL = XY.

Therefore, ∆KLM ≅ ∆XYZ.

**Note:** * Two triangles will be congruent only if they satisfy

any one of the four criterion mentioned above.

** Two triangles may not be congruent if any three parts

(elements) of one are equal to the corresponding parts of the other.

**Examples:**

(i) If two triangles have three angles of one equal to three

angles of the other, they are said to be equiangular. But equiangular triangles

need not be congruent.

Here, in the given figure, ∆KLM and ∆XYZ are

equiangular but not congruent.

In short, if two triangles are congruent, they must be

equiangular; but if they are equiangular, they may or may not be congruent.

(ii) If in two triangles,

two sides and one angle of one are equal to the corresponding sides and corresponding

angle of the other, the triangles need not be congruent.

In the adjoining figure, KL = XY, KM = XZ, ∠M

= ∠Z.

But ∆KLM and ∆XYZ are not congruent. For

congruency, two sides and the included angle of one must be equal to those of

the other.

**Note:** The abbreviation CPCTC is generally used for ‘Corresponding

parts of Congruent Triangles and Congruent’.

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