RASS Congruence — Proof Diagram Right-Angle–Side–Side in a Hilbert Plane Step 1 — Given triangles B A C AB AC B' A' C' ∠B = ∠B' = 90° AB ≅ A'B' (legs) AC ≅ A'C' (hypotenuses) Goal: △ABC ≅ △A'B'C' Step 2 — Transport A'B'C' onto ABC ℓ D B A C D is on ℓ: ∠ABD = 90° AD ≅ AC (transported hyp.) Need to show: C = D Step 3 — Contradiction if C ≠ D (case B * C * D shown) ℓ B C D A B * C * D ∠ACD 90° △ACD isoceles (AC ≅ AD) ⟹ ∠ACD ≅ ∠ADC (base angles, Prop 1.5) ∠ACB < 90° (angle sum in △ABC, since ∠ABC = 90°) ∠ACD = 180°− ∠ACB > 90° (supplementary angles at C) ∠ACD + ∠ADC > 180° contradicts Prop 1.17! Step 4 — Conclusion B C = D A Both cases C ≠ D (B*C*D and B*D*C) give contradictions. Therefore C = D, and the transported copy of △A'B'C' coincides with △ABC. ∴ △ABC ≅ △A'B'C' □ (No parallel postulate or continuity axiom used — holds in any Hilbert plane)