Right angles with regard to the latitude-longitude coordinate system, but not with regard to the surface of the earth. Recall that all meridians intersect at the poles.
@tantalor is technically correct; meridians and parallels intersect at right angles, even with respect to the (idealized) surface of the earth. Proof sketch: if the angles involved in a crossing were not all 90°, then either the crossing would not be symmetric about the meridian (it is), or one or both lines must have a sharp "kink" (i.e., have a non-smooth first derivative) at the intersection (neither does).
However, parallels, while smooth, are not straight, they "curl" toward the nearest pole (i.e., non-zero second derivative, relative to the earth's surface). This accounts for the non-"rectangular" shape of quadrangles and can be replicated on a 2D plane.
Take the lines tangent to the meridian & parallel at the point of intersection. Observe these lines are perpendicular. Hence the meridian & parallel meet at a right angle.
