In right triangles ABC and ABD with a common hypotenuse ab, legs BC and AB lie on parallel lines. Prove that AC is equal to BD.

By condition, the legs AC and ВD lie on parallel straight lines α and β, then the leg AB is parallel to the leg ВD.

The angle BAC is equal to the angle of the AВD, as the angles lying crosswise at the intersection of parallel straight lines AC and ВD secant AB.

Then, triangles ABC and AВD are equal in common hypotenuse AB and acute angle, and then AC = ВD, which was required to prove.