You figure it out

[drew]

The fact that it states:

Below the four parts are moved around

The partitions are exactly the same as those used above

Makes it hard for be to beleive that the angles are slightly different.

Would they just lie?

Has anyone who has printed it out, just cut one set of shapes and tried both triangles?

[I'm Murrin]

There's no contradiction: the individual pieces are the same in both shapes, but the two triangles--red and green--don't have the same slope. One triangle is 2/5, the other is 3/8.

[Kil Tyme]

Yup that's it. Slightly different slope; enough that in the bottom pic the slope difference, taken as a whole and extended over the length of the slope, allows for the extra space, represented by the "hole". Boy, that was not explained well, but at least my headache is gone.

[The Laughing Man]

Curry's Paradox: How Is It Possible?

[I'm Murrin]

The resolution of the apparent paradoxes is quite simple. The appearances notwithstanding, the three triangles involved have different angles so that their hypotenuses have different slopes. For n > 4, the discrepancy is imperceptible. But the cases n = 4 and n = 3 demonstrate this fact forcibly.

[The Laughing Man]

Curry himself has been interested in rearrangements that create holes entirely inside the resulting figure. But the variant with a square hole on the perimeter of the figure seems to me more popular nowadays.

it's not about what you want to see, Murrin, it's about what Paul Curry wants us to see. Do you see what I mean? He's a magician, not a mathmetician.