RHS Congruence Rule
The RHS (Right angle-Hypotenuse-Side) Congruence Rule applies specifically to right-angled triangles. If the hypotenuse and one side of a right triangle are equal to the hypotenuse and one side of another right triangle, the two triangles are congruent.
This rule is special because it works even though the equal angle (the right angle) is NOT between the two equal sides. For non-right triangles, SSA does not guarantee congruence — but for right triangles, it does.
What is RHS Congruence Rule - Grade 7 Maths (Congruence of Triangles)?
RHS Congruence Rule:
- If in two right-angled triangles, the hypotenuse and one side of one are equal to the hypotenuse and corresponding side of the other, the triangles are congruent.
Conditions:
- Both triangles must have a right angle.
- The hypotenuses must be equal.
- One leg (any one) must be equal.
RHS Congruence Rule Formula
If ∠B = ∠Q = 90°, AC = PR (hypotenuse), AB = PQ (one leg)
Then △ABC ≅ △PQR (by RHS)
Why it works: By Pythagoras Theorem, if the hypotenuse and one leg are known, the other leg is automatically determined (c² = a² + b² → b = √(c² − a²)). So all three sides are determined, making it equivalent to SSS.
Types and Properties
When to use RHS:
- Both triangles are right-angled.
- Hypotenuse and one leg are equal.
When NOT to use RHS:
- If neither triangle has a right angle — use SSS, SAS, or ASA instead.
- If both equal sides are legs (not hypotenuse) — this may require SAS instead.
Solved Examples
Example 1: Proving Congruence by RHS
Problem: In △ABC: ∠B = 90°, AC = 10 cm, AB = 6 cm. In △PQR: ∠Q = 90°, PR = 10 cm, PQ = 6 cm. Prove congruence.
Solution:
- ∠B = ∠Q = 90° (right angle)
- AC = PR = 10 cm (hypotenuse)
- AB = PQ = 6 cm (one leg)
- By RHS: △ABC ≅ △PQR
Answer: △ABC ≅ △PQR by RHS.
Example 2: Finding the Third Side
Problem: △ABC ≅ △XYZ by RHS. ∠B = 90°, AC = 13, AB = 5. Find BC.
Solution:
- By Pythagoras: BC² = AC² − AB² = 169 − 25 = 144
- BC = 12
Answer: BC = 12 (and YZ = 12 by CPCT).
Example 3: Word Problem — Shelves
Problem: Two right-angled triangular shelf brackets have the same hypotenuse (25 cm) and the same vertical side (7 cm). Are they identical?
Solution:
- Both are right triangles with equal hypotenuse and one equal side.
- By RHS, they are congruent.
Answer: Yes, the brackets are identical.
Real-World Applications
Real-world uses:
- Construction: Right-angled supports with equal hypotenuse and one leg are guaranteed identical.
- Manufacturing: Checking that right-triangular parts are identical.
- Geometry proofs: RHS is used to prove that perpendicular bisectors create congruent triangles.
Key Points to Remember
- RHS stands for Right angle-Hypotenuse-Side.
- Applies ONLY to right-angled triangles.
- If hypotenuse and one leg are equal, the triangles are congruent.
- It works because the third side is determined by Pythagoras Theorem.
- This is the only criterion where the equal angle need not be between the equal sides.
Practice Problems
- ∠A = ∠D = 90°, BC = EF = 15 cm, AC = DF = 9 cm. Prove △ABC ≅ △DEF.
- Can RHS be used for non-right triangles?
- Two right triangles: hypotenuse = 17, one leg = 8 each. Find the other leg.
- △PQR: ∠R = 90°, PQ = 20, QR = 12. △XYZ: ∠Z = 90°, XY = 20, YZ = 12. Are they congruent?
Frequently Asked Questions
Q1. What is RHS congruence?
If two right triangles have equal hypotenuse and one equal leg, they are congruent. RHS stands for Right angle-Hypotenuse-Side.
Q2. Why does RHS work but SSA does not?
In general SSA, there can be two different triangles with the same two sides and non-included angle. But in right triangles, the right angle fixes the third side via Pythagoras, so only one triangle is possible.
Q3. Can RHS be used for obtuse triangles?
No. RHS applies only to right-angled triangles. Both triangles must have a 90° angle.
Q4. Is RHS the same as SSS?
Not exactly, but RHS implies SSS. Once the hypotenuse and one leg are known in a right triangle, the third side is determined. So RHS is equivalent to SSS for right triangles.
