Do parallel, angle, triangle, area etc still apply in Mobius band? Normal geometry concepts, such as parallel, angle, area, triangle, do they still apply in Mobius band?
If not, in which case will they fail to do so?
For example, what would three lines on a Mobius band form? A triangle if not parallel? or it might be totally something else?
 A: Away from the edges, the geometry of a Moebius band is locally Euclidean, so the usual concepts and theorems of geometry all apply.  The issues involving edges are qualitatively no different than if you were trying to do geometry on a disk instead of a plane:  non-parallel lines may still never meet because they fall off the edge.
On the other hand, if you could form a one-sided surface with no boundaries, such as a Klein bottle, then things would be different.  For one thing, the sum of the angles in a triangle is no longer 180 degrees.  Since there is no obvious way to have a Klein bottle without some places being more tightly curved than others, this gets messy really fast, but the concepts are not a lot different than those of spherical geometry or hyperbolic geometry.
A: A priori a Mobius strip $S$ is a topological manifold, perhaps with boundary, depending on how we define it---for simplicity I'll assume no boundary.
If one endows $S$ with a Riemannian metric $g$---and there are many ways to do this---then one has available all of the usual trappings of Riemannian geometry. This includes measuring lengths of and angles between vectors, defining geodesics, distances between points, Gaussian curvature and so one, and as usual, these measurements and constructions depend heavily on the choice of $g$. Unlike most surfaces we work with, however, $S$ is nonorientable, so a metric does not determine even up to a fixed choice of sign a global choice of volume form on $S$, so we can define (unsigned) area, by integrating a Riemannian density rather than a volume form, but we cannot define a (globally consistent choice of) signed area.
One way to realize the Mobius strip is as the quotient of $\tilde S := \Bbb R \times (-1, 1)$ by the action $\Bbb Z \times \tilde S \to \tilde S$ defined by $n \cdot (x, y) := (x + n, (-1)^n y)$. In particular, $\Bbb Z$ acts by isometries of the usual Euclidean metric on $\tilde S$, which thus descends to a (locally) flat metric $\bar g$ on $S$.
Locally $(S, \bar g)$ behaves like any flat manifold, that is, like a patch of Euclidean space. Globally this metric behaves in some peculiar ways, however. For example, one can construct geodesics with an arbitrary number of self-intersections, which does not occur in (global) Euclidean space or on the (round) sphere. Some geodesics close but most do not. If one defines two geodesic to be parallel if they do not intersect, then the parallel postulate fails in general: Given a geodesic $\ell$ and a point $P$ not on the line, there may be zero, finitely many, or infinitely many lines through $P$ parallel to $\ell$. Depending on their relative positions, three geodesics may bound zero triangles, one, or many.
