Today our topic of discussion is Plane Geometry.
Plane Geometry
Plane Geometry
It has been mentioned earlier that point, straight line and plane are three fundamental concepts of geometry. Although it is not possible to define them properly, based on our real life experience we have ideas about them. As a concrete geometrical conception space is regarded as a set of points and straight lines and planes are considered the subsets of this universal set. That is,
Postulate 1: Space is a set of all points and plane and straight lines are the sub-sets of this set. From this postulate we observe that each of plane and straight line is a set and points are its elements. However, in geometrical description the notation of sets is usually avoided.
For example, a point included in a straight line or plane is expressed by the point lies on the straight line or plane’ or ‘the straight line or plane passes through the point!.
Similarly if a straight line is the subset of a plane, it is expressed by such sentences as ‘the straight line lies on the plane, or the plane passes through the straight line’. It is accepted as properties of straight line and plane that,
Postulate 2: For two different points there exists one and only one straight line, on which both the points lie.
Postulate 3: For three points which are not colinear, there exists one and only one plane, on which all the three points lie.
Postulate 4: A straight line passing through two different points on a plane lie completely in the plane.
Postulate 5:
1) Space contains more than one plane.
2) In each plane more than one straight lines lie.
3) The points on a straight line and the real numbers can be related in such a way that every point on the line corresponds to a unique real number and conversely every real number corresponds to a unique point of the line.
Remarks: The postulates from 1 to 5 are called incidence postulates. The concept of distance is also an elementary concept. It is assumed that.
Postulate 6:
1) Each pair of points P and Q determines a unique real number which is known as the distance between point P and Q and is denoted by PQ.
2) If P and Q are different points, the number PQ is positive. Otherwise, PQ = 0
3) The distance between P and Q and that between Q and P are the same, i.e. PQ = QP
Since PQ = QP this distance is called the distance between point P and point Q. In practical, this distance is measured by previously determined unit. According to postulate 5(c) one to one correspondence can be established between the set of points in every straight line and the set of real numbers. In this connection, it is admitted that.
Postulate 7: One-to-one correspondence can be established between the set of points in a straight line and the set of real numbers such that, for any points P, Q , PQ = |a – b| where, the one-to-one correspondence associates points P and Q to real numbers a and b respectively.
If the correspondence stated in this postulate is made, the line is said to have been reduced to a number line. If P corresponds to a in the number line, P is called the graph point of P and a the coordinates of P.
To convert a straight line into a number line the co-ordinates of two points are taken as 0 and 1 respectively. Thus a unit distance and the positive direction are fixed in the straight line. For this, it is also admitted that,
Postulate 8: Any straight line AB can be converted into a number line such that the coordinate of A is 0 and that of B is positive.
Remarks: Postulate 6 is known as distance postulate and Postulate 7 as ruler postulate and Postulate 8 as ruler placement postulate.
Geometrical figures are drawn to make geometrical description clear. The model of a point is drawn by a thin dot by tip of a pencil or pen on a paper. The model of a straight line is constructed by drawing a line along a ruler.
The arrows at ends of a line indicate that the line is extended both ways indefinitely. By postulate 2, two different points A and B define a unique straight line on which the two points lie. This line is called AB or BA line. By postulate 5(c) every such straight line contains infinite number of points.
According to postulate 5(a) more than one plane exist. There is infinite number of straight lines in every such plane. The branch of geometry that deals with points, lines lying in same plane and different geometrical entities related to them, is known as plane geometry.
In this textbook, plane geometry is the matter of our discussion. Hence, whenever something is not mentioned in particular, we will assume that all discussed points, lines etc lie in a plane.
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