Introduction to Euclid's Geometry

(i) False.
Through a single (one) point, infinitely many lines can pass. Euclid's postulate states that a unique line passes through *two* distinct points, not one. Hence the statement is false.

(ii) False.
By Euclid's first postulate, one and only one line can pass through two distinct points. Hence the statement is false.

(iii) True.
By Euclid's second postulate, a terminated line (line segment) can be produced indefinitely on both sides, giving an infinite straight line. Hence the statement is true.

(iv) True.
If two circles are equal, they can be superimposed on each other such that they coincide completely. Therefore their centres coincide and their radii must be equal. Hence the statement is true.

(v) True.
Given: $AB = PQ$ and $PQ = XY$.
By Euclid's first axiom — *Things which are equal to the same thing are equal to one another* — we conclude $AB = XY$. Hence the statement is true.

Before defining these terms, we need the following undefined/primitive terms: *point*, *line*, *plane*, *angle*, *distance*.

(i) Parallel Lines:
Two lines in the same plane are called parallel lines if they do not intersect each other at any point, i.e., they have no point in common.
- Terms needed first: *line*, *plane*, *point of intersection*.
- A *line* can be defined as a straight path extending infinitely in both directions.

(ii) Perpendicular Lines:
Two lines are called perpendicular lines if they intersect each other at a right angle ($90°$).
- Terms needed first: *line*, *point of intersection*, *angle*, *right angle*.
- A *right angle* is an angle whose measure is exactly $90°$.

(iii) Line Segment:
A line segment is a part of a line that is bounded by two distinct endpoints.
- Terms needed first: *line*, *point*.
- A *point* is that which has no part (position only).

(iv) Radius of a Circle:
The radius of a circle is the distance from the centre of the circle to any point on the circle.
- Terms needed first: *circle*, *centre*, *point*, *distance*.
- A *circle* is the set of all points in a plane that are at a fixed distance (radius) from a fixed point (centre).

(v) Square:
A square is a quadrilateral in which all four sides are equal in length and each interior angle is a right angle ($90°$).
- Terms needed first: *quadrilateral*, *side*, *angle*, *right angle*, *equal length*.
- A *quadrilateral* is a closed figure bounded by four line segments.

Undefined terms present:
Yes, both postulates contain undefined terms.
- In postulate (i): the terms *point* and *between* are undefined.
- In postulate (ii): the terms *point* and *line* (and the notion of a point lying *on* a line) are undefined.

Are the postulates consistent?
Yes, the two postulates are consistent. There is no contradiction between them.
- Postulate (i) says that between any two distinct points, a third point exists.
- Postulate (ii) says that not all points are collinear (at least three non-collinear points exist).
Both can be true simultaneously without any logical conflict.

Do they follow from Euclid's postulates?
No, these postulates do not directly follow from Euclid's five postulates.
- Postulate (i) is consistent with Euclid's postulates (since a line segment between two points can always be divided), but it is not explicitly stated by Euclid.
- Postulate (ii) is also consistent with Euclid's geometry (Euclid works in a plane with non-collinear points), but again it is not one of Euclid's original postulates.
Thus, they are independent assumptions that are compatible with, but not derived from, Euclid's postulates.

Given: Point C lies between points A and B such that $AC = BC$.

To Prove: $AC = \dfrac{1}{2}\,AB$

Figure: Place points on a line as: $A$——$C$——$B$

Proof:

Since point C lies between A and B, we have:
$$AC + CB = AB \quad \cdots (1)$$
(The whole equals the sum of its parts — Euclid's axiom: the whole is greater than the part; here $AC + CB$ makes up $AB$.)

It is given that:
$$AC = BC \quad \cdots (2)$$

Substituting (2) into (1):
$$AC + AC = AB$$
$$2\,AC = AB$$

Dividing both sides by 2 (using Euclid's axiom: things which are halves of the same things are equal to one another):
$$AC = \frac{1}{2}\,AB$$

Hence proved. $\blacksquare$