Introduction to Euclid’s Geometry
Basic Concepts and Important Points
1. Postulates: The basic facts which are taken, for granted, without proof and which are specific to geometry are called postulates.
2. Axioms: The basic facts which are taken for granted, without proof and which are used throughout in the mathematics are called axioms.
3. Theorem: The conclusions obtained through logical reasoning based on previously proved results and some axioms constitute a statement known as a theorem or a proposition.
4. Point: A point is represented by a fine dot made by a sharp pencil on a sheet of paper.
5. Plane: The surface of a smooth wall or the surface of a sheet of paper or the surface of a smooth black board are close examples of a plane.
6. Line: A line is breadthless length e.g.. if we fold a piece of paper, the crease in the paper represents a geometrical straight line. The edge of a ruler, the edge of the top of a table, the meeting place of two walls of a room are some examples of a geometrical straight line.
Collinear Points: Three or more points are said to be collinear, if there is a line which contains all of them.
Concurrent Lines: Three or more lines are said to be concurrent, if there is a point which lies on all of them.
Intersecting Lines: Two lines which meet at one point are said to be intersecting lines. The common point is called the ‘point of intersection’.
Note: Two distinct lines cannot have more than one point in common.
Parallel Lines: Two lines l and m in a plane are said to be parallel lines, if l ∩ m = ϕ. If l and m are two parallel lines in a plane, we can write l ‖ m.
Parallel Axiom: If l is a line and P is a point not on line l, there is one and only one line m which passes through P and is parallel to l.
Two lines which are both parallel to the same line, are parallel to each other.
If l, m, n are three lines in the same plane such that l intersects m and n ‖ m, then l intersects n also.
If land m are intersecting lines, l ‖ p and q ‖ m, then p and q also intersect.
If lines AB, AC, AD and AE are parallel to a line l, then points A, B, C, D and E are collinear.
Line Segment: In given two points A and B on a line l, the connected part (segment) of the line with end points at A and B, is called the line segment AB.
Interior Point of a Line Segment: A point P is called an interior point of a line segment AB, if P ε AB but P is neither A nor B.
Congruence of Line Segments: Two line segments AB and CD are congruent, if the trace-copy of one can be superposed on the other so as to cover it completely and exactly.
Line Segment Length Axiom: Every line segment has a length. It is measured in terms of ‘metre’ or Its parts.
Congruent Line Segment Length Axiom: Two congruent line segments have equal length and conversely, two line segments of equal length are congruent,
i.e., AB ≌ CD ⇔ l (AB) = l (CD).
Line Segment Addition Axiom: If C is any interior point of a line segment AB, then
Line Segment Construction Axiom: Given a point O on a line l and a positive real number r, there are exactly two points P1 and P2 on l, on either side of O such that
l (OP1) = l (OP2) = r cm.
Distance between Two Points: The distance between two points P and Q is the length of the line segment joining them and it is denoted by PQ.
25. Betweenness: Point C is said to lie between the two points A and B, if
(a) A, B and C are collinear points and
(n) AC + CB = AB.
Mid-point of a Line Segment: Given a line segment AB, a point M is said to be the mid-point of AB, if M is an interior point of AB such that AM = MB.
Line through M, other than line AB is called the bisector of the segment AB.
Opposite Rays: Two rays AB and AC are said to be opposite rays if they are collinear and point A is the only common point of these two rays.
Note: Two rays or two line segments or a line segment and a ray (line) are said to be parallel, if the lines containing them are parallel.
4. Undefined Terms: There are some terms like point etc which have no definitions, these terms are called undefined terms.
In geometry, a point, a line and a plane (in Euclid’s words, a plane surface) are undefined terms.
5. Euclid’s Axioms
- Things which are equal to the same thing are equal to one another.
- If equals are added to equals, the wholes are equal.
- If equals are subtracted from equals, the remainders are equal.
- Things which coincide with one another are equal to one another.
- The whole is greater than the part.
- Things which are double of the same things are equal to one another.
- Things which are halves of the same things are equal to one another.
6. Euclid’s Five Postulates: Euclid’s five postulates are given below
Postulate 1: A straight line may be drawn from any one point to any other point.
Axiom Given two distinct points, there is a unique line that passes through them.
Postulate 2: A terminated line can be produced indefinitely. Euclid’s terminated line is called a line segment. So, according to the present day terms, ‘A line segment can be extended on either side to form a line’.
Postulate 3: A circle can be drawn with any centre and any radius.
Postulate 4: All right angles are equal to one another.If ∠1 = 90°
and ∠2 = 90°
Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
e.g., The line PQ falls on lines AB and CD such that the sum of the interior angles 1 and 2 is less than 180° on the left side of PQ. Therefore, the lines AB and CD will eventually intersect on the left side of PQ.
Note: Two distinct lines cannot have more than one point in common.
7. Equivalent Versions of Euclid’s Fifth Postulate: ‘For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to ‘l’.
This result can also be stated in the following form.
Two distinct intersecting lines cannot be parallel to the same line.
