RHS Congruence Rule

The RHS Congruence Rule is a special congruence test used only for right-angled triangles.

It states that if:

• the right angle,
• the hypotenuse, and
• one corresponding side

of one right triangle are equal to the corresponding right angle, hypotenuse, and side of another right triangle, then the two triangles are congruent.

 Right angle + Hypotenuse + one side → Congruence

RHS (Right angle–Hypotenuse–Side) Rule:

If in two right-angled triangles:

• their right angles are equal,
• their hypotenuses are equal, and
• one pair of corresponding legs are equal,

then the triangles are congruent.

Symbolically, for triangles \(\triangle ABC\) and \(\triangle DEF\), right-angled at \(B\) and \(E\):

• \(AB = DE\)
• \(AC = DF\) (hypotenuses)
• \(\angle B = \angle E = 90^\circ\)

Then:

\(\triangle ABC \cong \triangle DEF\)

A right-angled triangle is completely determined when:

• one acute angle,
• or the hypotenuse and one leg

are known. So if two right triangles have the same hypotenuse and one matching leg, their shape and size become fixed.

This makes RHS a reliable congruence test.

In a right triangle:

• The hypotenuse is the side opposite the right angle and is the longest side.
• The other two sides are called the legs of the triangle.

   A                     D
   |\                    |\
   | \                   | \
 b |  \ c           b    |  \ c
   |   \                 |   \
   B----C                E----F
      a (hyp.)              a (hyp.)
Right angle at B and E → Congruent

The RHS rule is helpful when dealing with right-angled components, such as:

• triangular supports in construction,
• ramps and stairway geometry,
• measuring heights using right angles,
• structural designs requiring precise right triangles.

RHS is widely used because right triangles appear very often in practical applications.