Cells must be in osmotic equilibrium with their surrounding environments, because if they swell or shrink their membranes will rupture. what is the correct answer

Well I'm not exactly certain where the teacher is going with this, but an often used example is red blood cells (RBCs) aka: erythrocytes.
RBCs are suspended in blood plasma as they flood through vessels around and around the body, so the osmolarity (amount of small particles that affect osmosis) must remain relatively constant. This is termed "isotonic", meaning the same amount of osmosis-influencing particles that are there inside the RBCs' cytosol, within their plasma membranes.
If the plasma osmolarity get too high, called hypertonic (as with extra salt particles) then water inside the RBCs will have an osmotic force driving it out of the cells' membranes, to flow where there are more salt particles. This will lead to cell shrinkage (called "crenation").
Counter to that, if the plasma osmolarity gets too low, as due to low plasma salt with excessive water intake (for example from the condition "water intoxication"), then the plasma will be hypotonic with respect to the intracellular cytosol concentration. This can result in water rushing into the RBCs' membranes via osmosis, causing the cells to swell from discs into spheres (balls), or even rupture and burst (a phenomenon called "hemolysis").

HOPE THOSE EXAMPLES HELP!!

Animal Cells is the Correct Answer

The answer is constant.