State whether the following statements are true (T) or false (F):

(i) The line segment $\overline{AB}$ is the shortest route from A to B.

(ii) A line cannot be drawn wholly on a sheet of paper.

(iii) A line segment is made of infinite (uncountable) number of points.

(iv) Two lines in a plane always intersect.

(v) Through a given point only one line can be drawn.

(vi) Two different lines can be drawn passing through two distinct points.

(vii) Every simple closed curve is a polygon.

(viii) Every polygon has atleast three sides.

(ix) A vertex of a quadrilateral lies in its interior.

(x) A line segment with its end-points lying on a circle is called a diameter of the circle.

(xi) Diameter is the longest chord of the circle.

(xii) The end-points of a diameter of a circle divide the circle into two parts; each part is called a semicircle.

(xiii) A diameter of a circle divides the circular region into two parts; each part is called a semicircular region.

(xiv) The diameters of a circle are concurrent. The centre of the circle is the point common to all diameters.

(xv) Every circle has unique centre and it lies inside the circle.

(xvi) Every circle has unique diameter.

(i) True — The shortest distance between two points is the straight line segment joining them.

(ii) True — A line extends indefinitely in both directions, so only a part of it can be shown on a sheet of paper.

(iii) True — Between any two points of a line segment, infinitely many points can be found.

(iv) False — Two parallel lines in a plane never intersect.

(v) False — Infinitely many lines can be drawn through a single given point.

(vi) False — Only one unique line can be drawn passing through two distinct points.

(vii) False — A simple closed curve is a polygon only if it is made up entirely of line segments. A circle, for example, is a simple closed curve but not a polygon.

(viii) True — The polygon with the least number of sides is a triangle, which has three sides.

(ix) False — A vertex of a quadrilateral lies on its boundary, not in its interior.

(x) False — A line segment with its end points on a circle is called a chord. It is called a diameter only when it also passes through the centre of the circle.

(xi) True — The diameter is the longest chord of the circle because it passes through the centre.

(xii) True — A diameter divides the circle into two equal arcs, each called a semicircle.

(xiii) True — A diameter divides the circular region into two equal parts, each called a semicircular region.

(xiv) True — All diameters of a circle pass through its centre, so they are concurrent at the centre.

(xv) True — A circle has a unique centre and the centre lies in the interior of the circle.

(xvi) False — A circle has infinitely many diameters (any chord passing through the centre is a diameter).