Congruence Among Right-Angled Triangles

RHS Criterion (Right angle-Hypotenuse –Side)

This criterion says that the two right-angled triangles will be congruent if the hypotenuse and one side of one triangle are equal to the corresponding hypotenuse and one side of another triangle.

If Right angle ∠B = ∠E

Hypotenuse AC = DF

Side BC = EF

Then, ∆ABC ≅ ∆DEF

Example

Prove that ∆RSV ≅ ∆RKV, if RS = RK = 7 cm and RV = 5 cm and is perpendicular to SK.

Solution

In ∆RSV and ∆RKV,

Given

RS = RK = 7 cm

RV = RV = 5 cm (common side)

If RV is perpendicular to SK then

 ∠RVS = ∠RVK = 90°.

Hence, ∆RSV ≅ ∆RKV

As in the two right-angled triangles, the length of the hypotenuse and one side of both the sides are equal.

Remark: AAA is not the criterion for the congruent triangles because if all the angles of two triangles are equal then it is not compulsory that their sides are also equal. One of the triangles could be greater in size than the other triangle.

In the above figure, the two triangles have equal angles but their length of sides is not equal so they are not congruent triangles.

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