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I was taught in class that in case of perpendicular axis theorem $I_z=I_x+I_y$ (therefore,$I_x=I_z-I_y$, according to me) but in the next class my teacher told me that $I_x=I_z+I_y$.

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As Wikipedia states:

The perpendicular axis theorem states that the moment of inertia of a plane lamina about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia of the lamina about the two axes at right angles to each other, in its own plane intersecting each other at the point where the perpendicular axis passes through it.

Here the z-axis is supposed to be the axis perpendicular to the plane of the lamina (and not something else). So as it turns out, you are right (as per the unsaid convention). Note that he/she uses x-axis as the axis perpendicular to the plane of the lamina and you should not equate both.

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The perpendicular axis theorem actually has two statements or necessities if you must say: $$ $$ 1) The rigid body to be considered should be two dimensional $$ $$ 2) If there exist three mutually perpendicular axis I1, I2, I3, out of which any two(lets say I1 and I2) are in the plane of the 2D body then: $$ I1 + I2 = I3 $$ Now out of I1,I2,I3 which corresponds to x,y or z is completely arbitary and your choice. Infact in any coordinate system assigning xyz to an axis is your choice, the thing is you should be aware of your axes, because after all its the logic that should work out. I believe the definition told to you was more of mathematical and typical rather than logical and generalised. You should definitely discuss with your teacher.

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