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. $\blacksquare$

To Prove: Every line segment has one and only one mid-point.

Proof by contradiction:

Let $AB$ be a line segment. Assume, if possible, that $AB$ has two mid-points, say $C$ and $D$ (where $C \neq D$).

Since $C$ is a mid-point of $AB$:
$$AC = CB \quad \Rightarrow \quad AC + AC = AC + CB \quad \Rightarrow \quad 2\,AC = AB \quad \cdots (1)$$

Since $D$ is a mid-point of $AB$:
$$AD = DB \quad \Rightarrow \quad AD + AD = AD + DB \quad \Rightarrow \quad 2\,AD = AB \quad \cdots (2)$$

From (1) and (2), using Euclid's axiom (*things which are equal to the same thing are equal to one another*):
$$2\,AC = 2\,AD$$

Dividing both sides by 2 (Euclid's axiom: halves of equal things are equal):
$$AC = AD$$

This means $C$ and $D$ are the same point, which contradicts our assumption that $C \neq D$.

Therefore, our assumption is wrong, and every line segment has one and only one mid-point.

Hence proved. $\blacksquare$

Given: Points A, B, C, D lie on a line in the order A—B—C—D, and $AC = BD$.

To Prove: $AB = CD$

Proof:

From the figure, point B lies between A and C, so:
$$AC = AB + BC \quad \cdots (1)$$

Point C lies between B and D, so:
$$BD = BC + CD \quad \cdots (2)$$

It is given that:
$$AC = BD \quad \cdots (3)$$

Substituting (1) and (2) into (3):
$$AB + BC = BC + CD$$

Subtracting $BC$ from both sides (using Euclid's axiom: *if equals are subtracted from equals, the remainders are equal*):
$$AB = CD$$

Hence proved. $\blacksquare$

Euclid's Axiom 5 states: *"The whole is greater than the part."*

This axiom is considered a universal truth because:

1. It is self-evident and needs no proof. It is intuitively obvious in every situation — if you take a part of any quantity, the original whole must be larger than that part.

2. It applies universally — not just to geometry, but to all measurable quantities in the real world. For example:
- If a number $x$ is a part of a number $y$, then y > x.
- If a line segment $PQ$ is a part of line segment $AB$, then AB > PQ.
- If a region (area) is a part of a larger region, the larger region has greater area.

3. It holds true in every context without exception, making it a universal truth applicable across mathematics and everyday life.

Thus, because this axiom is obviously true in all cases and requires no proof or special conditions, it is accepted as a universal truth.