Triangles [Congruency in Triangles]

210. The RHS Rule · Hypotenuse & side guarantee congruence

If two right triangles have equal hypotenuse and one leg, they are always congruent.

By the RHS rule (Right angle–Hypotenuse–Side), if the hypotenuse and one side of a right-angled triangle equal those of another, the triangles are congruent. The right angle with the hypotenuse and one leg fixes the third side through Pythagoras, so the shape is locked. RHS applies only to right-angled triangles.

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Selina ICSE: Triangles [Congruency in Triangles]

What this lesson covers

Try to break it

Drag C along the horizontal base. Both triangles' bases stretch together, keeping the hypotenuse and the right-angle leg matched. Try to drag C to a length where one triangle differs from the other — you can't, the handle drives both. RHS holds.

How you build it

Construct two right triangles with equal hypotenuse and one leg to test the RHS rule.

Point tool: mark point B — the right-angle corner.
Point tool: mark point A above B — BA is the given leg.
Segment tool: join B to A. This is one leg.
Perp tool: click B (on leg BA), then click to the side — this raises the right angle at B; the base BC lies on this perpendicular.
Point tool: mark point C on the perpendicular, a little way out from B. You pick the spot — BC is the second leg and AC will be the hypotenuse you copy later.
Now the second triangle. Point tool: mark point E to the right — its right-angle corner.
Ray tool: click E, then click upward — leg ED will lie on this ray.
Arc tool: click centre B, drag until the arc snaps to A. This locks the radius to the leg BA.
Arc tool: click centre E — with BA locked, the arc crosses the leg ray at D, so ED = BA.
Point tool: mark D where the arc meets the leg ray.
Segment tool: join E to D.
Perp tool: click E (on leg ED), then click to the side to raise the right angle at E.
Arc tool: click centre A, drag until the arc snaps to C. This locks the radius to the hypotenuse AC.
Arc tool: click centre D — with AC locked, the arc crosses the perpendicular at F, so DF = AC.
Point tool: mark F where the arc meets the perpendicular.
Segment tool: join D to F. Right angle at E, leg ED = BA, hypotenuse DF = AC — so triangle DEF is congruent to ABC by RHS.

The proof, step by step

Prove that an equal hypotenuse and one leg make right triangles congruent (RHS).

∠B = ∠E = 90° (Given)
AC = DF (Hypotenuse, Given)
AB = DE (Side, Given)
∴ ΔABC ≅ ΔDEF (By RHS Congruence Rule)

Worked example

In right ΔPQR and ΔXYZ, ∠Q = ∠Y = 90°. PR = 13 cm, PQ = 5 cm, and XY = 5 cm. If ΔPQR ≅ ΔXYZ by RHS rule, what is the length of YZ?

Since ΔPQR ≅ ΔXYZ by RHS, corresponding sides are equal. PQ = XY = 5 cm. PR = XZ = 13 cm. By Pythagoras theorem, YZ = √(XZ² - XY²) = √(169 - 25) = √144 = 12 cm.

5 cm
12 cm — correct
13 cm
8 cm