Does Pythagoras theorem hold for length contraction according to Special relativity? Here is a thought experiment.
Consider a frame S in which an observer is at rest and she looks ahead to find a right angle triangle of sides 3 (x-axis),4(y-axis) and hypotenuse 5. Now consider another frame S' in which is moving at a speed 'v' moving along x-axis, with respect to frame S, that has another observer.
Will the observer in the frame S' be able to accurately deduce the Pythagoras theorem or will length contraction along the x-axis lead to some other x-axis length for the triangle?
 A: I have made a diagram to clarify the situation
black is for the $S$ coordinate system
red is for the $S'$ coordinate system with motion $v'$ relative to $S$
The black triangle is what $S$ sees. The red triangle with smaller reduced base $b'$ is what $S'$ sees. $S$ and $S'$ will both see right triangle and both will obey the Pythagorean Theorem.
$$
a^2+b^2=c^2
$$
and also
$$
a'^2+b'^2=c'^2
$$
In fact they will also both know what each other sees by using the Length Contraction formulas.
I hope that helps clarify your question.

A: In all inertial frames the metric is $ds^2=-c^2 dt^2 + dx^2 + dy^2 + dz^2$. For any moment in time we can set $dt=0$ and recover the Pythagorean theorem.
So the Pythagorean theorem holds for space in all inertial frames in special relativity. So, if one leg of a right triangle is length contracted, then you can use the Pythagorean theorem to determine the contraction of the hypotenuse.
A: As an alternative to the other answers, you could just use the principle of relativity rather than any explicit form of the Lorentz transformation (which is deduced from it): this says that the laws of physics can be formulated in a way that is the same in all inertial frames.
This implies that there is no experiment that would tell you your state of (uniform) motion, in particular if the Pythagorean theorem holds in one frame, it holds in all frames.
A: Your thought experiment needs to start with a more basic questions: "What is a straight line?" and "What is a right angle?" and "How to measure distance?"
If we are doing real physics experiment, and we say use a beam of laser as a "straight line" and an atomic clock to measure the time of transit between points thus measuring distance, and a mirror to reflect laser in a right angle. And you put the whole setup in a spacecraft traveling at near light speed.
From the observer inside the spacecraft, obviously Pythagoras Theorem holds.
For a stationary observer outside the spacecraft, with the Lorenz transform, I believe Pythagoras Theorem still holds.
However if the spacecraft fly next to a black hole, the space is not Cartesian, thus Pythagoras Theorem does not hold. (Same applies to accelerating frame.)
A: If you draw a right triangle on a sheet of rubber, and you uniformly stretch the rubber in a direction aligned with one of the non-hypotenuse sides...you still have a right triangle. However, the angle has changed. Do you suppose there could be any parallels with your thought experiment?
