Congruency: Congruent Triangles
129. RHS Congruence Criterion · Right angle, Hypotenuse, Side
If hypotenuse and one side match, the triangles are congruent.
RHS congruence test (Right-angle Hypotenuse Side): two right triangles are congruent if their hypotenuses are equal AND one pair of legs is equal. The third side and remaining angles are then determined by Pythagoras.
What this lesson covers
Try to break it
Drag E along the base. The blue right triangle stretches, but its hypotenuse and one leg stay locked equal to the red triangle's. Try to make the two triangles non-congruent — impossible. RHS holds.
How you build it
Construct two right triangles by RHS.
- Draw a line segment AB, the first leg of the right triangle.
- At B, draw a line perpendicular to AB.
- Mark point C on the perpendicular so that BC equals the given length.
- Draw segment AC to form the hypotenuse of the first triangle.
- Draw a line segment DE, the first leg of the second right triangle.
- At E, draw a line perpendicular to DE.
- Mark point F on the perpendicular so that EF equals BC.
- Draw segment DF to form the hypotenuse of the second triangle.
The proof, step by step
Prove that an equal hypotenuse and one side make right triangles congruent (RHS).
- In ΔABC and ΔFED, ∠B = ∠E = 90° (Given)
- AC = FD (Given, hypotenuse)
- AB = FE (Given, side)
- ∴ ΔABC ≅ ΔFED (by RHS Congruence Rule)
Worked example
In ΔABC and ΔDEF, ∠B = ∠E = 90°, AC = DF = 10 cm, and BC = EF = 6 cm. Which congruence criterion proves ΔABC ≅ ΔDEF?
Both triangles are right-angled at B and E. The hypotenuses (AC and DF) are equal, and one pair of corresponding legs (BC and EF) are equal. Therefore, ΔABC ≅ ΔDEF by the RHS congruence criterion.
- SAS
- ASA
- RHS — correct
- SSS