Question

Explain RHS test of congruence of triangles.​

Answer 1

RHS Congruence Rule
Theorem: In two right-angled triangles, if the length of the hypotenuse and one side of one triangle, is equal to the length of the hypotenuse and corresponding side of the other triangle, then the two triangles are congruent.

In the following figure, AB = BC and AD = CD. Show that BD bisects AC at right angles.

Congruence Of Triangles

Solution: We are required to prove ∠BEA = ∠BEC = 90° and AE = EC.

Consider ∆ABD and ∆CBD,

AB = BC (Given)

AD = CD (Given)

BD = BD (Common)

Therefore, ∆ABD ≅ ∆CBD (By SSS congruency)

∠ABD = ∠CBD (By CPCT)

Now, consider ∆ABE and ∆CBE,

AB = BC (Given)

∠ABD = ∠CBD (Proved above)

BE = BE (Common)

Therefore, ∆ABE≅ ∆CBE (By SAS congruency)

∠BEA = ∠BEC (CPCTC)

And ∠BEA +∠BEC = 180° (Linear pair)

2∠BEA = 180° (∠BEA = ∠BEC)

∠BEA = 180°/2 = 90° = ∠BEC

AE = EC (CPCTC)

Hence, BD is a perpendicular bisector of AC.

Hope, it helps you..

Answer 2

The test is therefore given the initials RHS for 'Right angle', 'Hypotenuse, 'Side'. The hypotenuse and one side of one right-angled triangle are respectively equal to the hypotenuse and one other side of another right-angled triangle then the two triangles are congruent.