Given four points A B C and D no three of which are collinea

Given four points, A, B, C, and D, no three of which are collinear and such that any pair of the segments AB, BC, CD, and DA either have no point in common or have only an endpoint in common. We can then define the quadrilateral DABCD to consist of the four segments mentioned, which are called its sides, the four points being called its vertices (see Figure 1.21). (Note that the order in which the letters are written is essential. For example, DABCD may not denote a quadrilateral because, for example, AB might cross CD. If ?ABCD did denote a quadrilateral, it would not denote the same one as DACDB. Which permutations of the four letters A, B, C, and D do denote the same quadrilateral as DABCD?) Using this definition, define the following notions: The angles of DABCD. Adjacent sides of DABCD. Opposite sides of DABCD. The diagonals of DABCD. A parallelogram. (Use the word \"parallel.\")

Solution

(a) An angle of Quadrilateral ABCD is the angle formed by 2 segment which have only one endpoint in common. Thus, if AB and AD are 2 line segment of Quadrilateral ABCD and these have only vertex A in common and also they do not cross each other, then angle formed by these 2 line segments is an angle of Quadrilateral ABCD. The set of all such angles represent all angles of Quadrilateral ABCD. Therefore, angles are : formed by : (AB, AD) i.e angle A, (BA, BC) angle B, (BC,CD) angle C, (DC,DA) angle D.

(b) 2 sides or segments of Quadrilateral ABCD which have only an endpoint in common and which do not cross each other, constitute a pair of adjacent sides of Quadrilateral ABCD. So adjacent sides of Quadrilateral ABCD are (AB, BC), (BC,CD), (CD,DA), (DA,AB).

(c) Opposides of Quadrilateral ABCD are those sides or segements which have no point in common. So AB and CD are opposite sides and BC and DA are opposite sides.

(d) Diagonals of Quadrilateral ABCD are defined as if we take any two adjacent sides of Quadrilateral ABCD and join their second endpoint by a line segment, then this line segment is called a diagonal of Quadrilateral ABCD. So AC and BD are diagonals of Quadrilateral ABCD.

(e) If both the pairs of opposite sides (i.e. AB & CD and BC & DA) are parallel to each other (i.e. AB || CD and BC || DA) then the Quadrilateral ABCD is called as Parallelogram.