Can glass be both fragile and brittle? I was told that the glass which is used in construction is brittle, not fragile.

Fragility is a non-technical term referring intuitively to how easily something might break. Brittleness incorporates a notion of fragility, but mainly refers to how something breaks.
Often when people say something is "brittle", they mean it is prone to brittle fracture (like glass or rock or cast iron) rather than plastic deformation (like polythene, ductile metals, wood). So one might say that cast iron is brittle but not fragile, because it fails in a brittle mode, but requires a large, well aimed knock or shock to cause it to fail, for example dropping it onto a hard surface from a sufficient height causes it to fracture into shards. If it was fragile and brittle, then a drop onto a hard surface from a very low height would cause sufficient shock to break it, or I could fracture it by bending it with my bare hands. On the other hand, a thin film of plastic or wood or old cloth can be fragile, but not brittle, in that it is easily broken, but it fails through plastic deformation, ie pushing on it continuously it deforms and then tears, but a shock such as dropping it to the ground does not cause it to break into shards.
Brittle not fragile: I can lean on a window without falling through, but if I drop it, it will fracture into shards.
Fragile not brittle: I can drop a piece of paper without damage, but leaning on paper stretched across a frame I can easily put my hand through it.
