but why one half shaded square versus the other?
The dark stripe pattern on the right in the square was matched by the right dark stripe pattern found in the oval and circle. Therefore the square with the dark stripe in its right interior is the missing proper shape!
The key was the orientation patterns on the inside of the geometric shapes....you looked for what was missing...in this case a square with the dark stripe in the right side of its interior. (The oval and circle with the dark stripe in the right interior were already shown)