I'm not sure if my geometry proof of Hypotenuse/Leg congruence is correct

[jdinatale]

I'm obviously not allowed to use Angle-Side-Side or the Pythagorean Theorem. I can only use Euclid's 5 postulates and the theorems that follow(Or the equivalent SMSG postulates and theorems, Hilbert postulates, etc.).

I think my usage of correct vocabulary is wrong, or maybe my whole proof is.

1. The Problem: If the hypotenuse and leg of one right triangle are congruent, respectively, to the hypotenuse and leg of another, then the two triangles are congruent.

The Attempt at a Solution

 

[LCKurtz]

I'm no expert on what you can and can't do with Euclid axioms, but I think "moving" the triangle so the edges abut like that is a bit iffy. But I think your proof can be made rigorous like this. In your original picture, at P construct angle QPC' equal to angle BAC and swing a compass arc from PR to PC' to make the lengths equal. Now your triangle QPC' is congruent to your original BAC by side-angle-side. Then you are home free, I think.

 

[mathwonk]

i think you are fine. It just depends on what you take for granted. euclid himself moved triangles in his proof of SAS congruence. Then one gets also SSS congruence using moving, and then one can copy triangles by copying their sides, hence can construct angles. later hilbert suggested giving yourself instead SAS and the ability to construct or copy angles.

These are just different versions of the same proof. But to be rigorous you should say what your axioms are. if all you have is euclid's original 5, it is hard to do much, e.g. you can't even copy angles.

If you want to use more, but stuff still contained in euclid, just apply pythagoras to get the other sides congruent as well..

they may be reasonably augmented to include the ones he used without mention, such as existence of rigid motions, plane separation property by lines, and intersection properties of circles. see hilbert's foundations, or hartshorne's book, or the epsilon camp notes on my web page at UGA.

 

[mathwonk]

by the way have you noticed there is a SSA theorem if the angle is either a right angle or an obtuse angle?