(i) Only one line can pass through a single point.
False. Reason: There are an infinite number of lines which can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
False. Reason: Only one unique line passes through two distinct points (Axiom 5.1).
(iii) A terminated line can be produced indefinitely on both the sides.
True. Reason: According to Euclid’s Postulate 2, a terminated line can be produced indefinitely.
(iv) If two circles are equal, then their radii are equal.
True. Reason: If two circles are equal, they coincide with each other, so their centres and radii must be equal (by the principle of superposition or Axiom 4).
(v) In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.
True. Reason: According to Euclid’s Axiom 1, things which are equal to the same thing are equal to one another.
(i) Parallel lines
Definition: Two distinct lines in a plane that do not intersect anywhere are called parallel lines.
Undefined terms: Point, line, plane.
Point: That which has no part. Line: Breadthless length. Plane: A flat surface extending indefinitely in all directions.
(ii) Perpendicular lines
Definition: Two lines in a plane are said to be perpendicular if they intersect each other at right angles (90°).
Undefined terms: Line, plane, right angle.
Right angle: An angle equal to 90°.
(iii) Line segment
Definition: A line segment is a part of a line that is bounded by two distinct end points.
Undefined terms: Point, line.
(iv) Radius of a circle
Definition: The radius of a circle is the distance from the centre of the circle to any point on its circumference.
Undefined terms: Point, circle, centre, circumference.
Circle: A set of points at a fixed distance from a fixed point (centre).
(v) Square
Definition: A square is a quadrilateral with all sides equal and all angles equal to 90°.
Undefined terms: Point, line, angle, quadrilateral.
Quadrilateral: A closed figure bounded by four line segments.
Undefined terms: Point, between, line.
These postulates contain undefined terms like 'between' and 'on the same line'.
They are consistent because they deal with different situations: one assumes a point between two points, the other assumes points not collinear.
They do not follow directly from Euclid’s postulates, as Euclid’s postulates are about lines, circles, and angles, but these are additional assumptions about points.
Given: Point C lies between A and B, and AC = BC.
By Euclid’s Axiom 4, things which coincide with one another are equal.
AB coincides with AC + CB, so AB = AC + CB.
Since AC = BC, AB = AC + AC = 2 AC.
Therefore, AC = 1/2 AB.
Figure: A---C---B, with AC = BC.
Assume there are two mid-points C and D on AB.
Then AC = 1/2 AB and AD = 1/2 AB.
By Axiom 1, AC = AD.
But if C and D are different, this would contradict unless C coincides with D.
Hence, only one mid-point.
(Using the uniqueness from Axiom 5.1 implicitly.)
Assuming order A-B-C-D on a line.
AC = AB + BC, BD = BC + CD.
Given AC = BD, so AB + BC = BC + CD.
By Axiom 3, subtract BC from both sides: AB = CD.
Axiom 5: The whole is greater than the part.
This is a universal truth because it holds not just in geometry but in everyday life and other fields.
For example, a whole cake is greater than half a cake; total salary is greater than expenditure, etc.
It is an obvious truth applicable universally.
Given: Points A, B, C on a line with B between A and C.
AC coincides with AB + BC.
By Euclid’s Axiom 4, things which coincide are equal.
Therefore, AB + BC = AC.
Given: Line segment AB.
Draw a circle with centre A and radius AB.
Draw another circle with centre B and radius AB.
The circles intersect at point C (assuming they meet).
Then, AB = AC (radii of first circle), AB = BC (radii of second circle).
By Axiom 1, AB = AC = BC, so triangle ABC is equilateral.
Proof: Assume two lines l and m intersect at two points P and Q.
Then, two distinct points P and Q have two lines passing through them.
This contradicts Axiom 5.1 that only one unique line passes through two distinct points.
Hence, two distinct lines cannot have more than one point in common.
1. Though Euclid defined a point, a line, and a plane, the definitions are not accepted by mathematicians. Therefore, these terms are now taken as undefined.
2. Axioms or postulates are the assumptions which are obvious universal truths. They are not proved.
3. Theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning.
4. Some of Euclid’s axioms were :
(1) Things which are equal to the same thing are equal to one another.
(2) If equals are added to equals, the wholes are equal.
(3) If equals are subtracted from equals, the remainders are equal.
(4) Things which coincide with one another are equal to one another.
(5) The whole is greater than the part.
(6) Things which are double of the same things are equal to one another.
(7) Things which are halves of the same things are equal to one another.
5. Euclid’s postulates were :
Postulate 1 : A straight line may be drawn from any one point to any other point.
Postulate 2 : A terminated line can be produced indefinitely.
Postulate 3 : A circle can be drawn with any centre and any radius.
Postulate 4 : All right angles are equal to one another.
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.