Introduction to Euclid's Geometry — NCERT Solutions, AI Tutor & Practice

(i) False. Through a single point we can draw infinitely many lines in different directions. So it is not true that only one line passes through a single point.

(ii) False. By Euclid's first postulate, exactly one line can be drawn through two distinct points, not infinitely many.

(iii) True. A terminated line (a line segment) can be extended endlessly in both directions, according to Euclid's second postulate.

For each definition we first need certain undefined/earlier terms such as point, line, plane, distance and angle.

(i) Parallel lines: Two lines in the same plane that never meet, i.e. the distance between them is always the same. (Needs line, plane, distance.)

(ii) Perpendicular lines: Two lines that intersect such that the angle between them is $90^\circ$. (Needs line, intersect, angle.)

(iii) Line segment: The part of a line lying between two given points, including those two end points. (Needs line, point.)

Yes — in each case some terms must be defined earlier, which is why Euclid began with undefined terms.

(i) Axiom. 'The whole is greater than the part' is one of Euclid's common notions (axioms) that applies to all mathematics, not just geometry.

(ii) Postulate. This is Euclid's first postulate, an assumption specific to geometry.

(iii) Theorem. The angle-sum property of a triangle is not assumed — it is proved using postulates and axioms, so it is a theorem.

Key idea: Axioms/common notions are universal self-evident truths, postulates are geometric assumptions, and theorems are proved results.

Axiom (common notion): A basic self-evident truth assumed to be true throughout mathematics, not restricted to geometry. Example: 'Things which are equal to the same thing are equal to one another.'

Postulate: An assumption specific to geometry that Euclid stated without proof to build the subject. Example: 'A straight line can be drawn joining any two points.' (First postulate.)

Difference: Axioms are general assumptions used everywhere in mathematics, while postulates are geometry-specific assumptions.

(a) Through one given point: Infinitely many lines can pass through a single point, since a line can be drawn in any direction through that point.

(b) Through two distinct points: Exactly one line can pass through two distinct points. This is guaranteed by Euclid's first postulate, which states that a unique straight line joins any two points.

Euclid's fifth postulate is complicated in its original form. A simpler equivalent statement is Playfair's Axiom:

'For every line $\ell$ and for every point $P$ not lying on $\ell$, there exists a unique line through $P$ that is parallel to $\ell$.'

In other words, through a point outside a given line, exactly one parallel line can be drawn. This version is logically equivalent to Euclid's fifth postulate but far easier to state and use.

Since $C$ lies between $A$ and $B$, by the betweenness property: $AC + BC = AB.$

We are given that $AC = BC$. Substituting $BC = AC$: $AC + AC = AB$ $2\,AC = AB.$

Using Euclid's axiom (equals divided by equals are equal), divide both sides by $2$: $AC = \tfrac{1}{2}\,AB.$

Hence proved. Such a point $C$ is called the midpoint of $AB$.

Let $AB$ be a line segment and suppose it has two midpoints $C$ and $D$.

Since $C$ is a midpoint: $AC = \tfrac{1}{2}\,AB. \quad (1)$ Since $D$ is a midpoint: $AD = \tfrac{1}{2}\,AB. \quad (2)$

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

But both $C$ and $D$ lie on the same segment $AB$ and are measured the same distance from $A$. Two points at the same distance from $A$ along $AB$ must coincide, so: $C = D.$

This contradicts our assumption of two distinct midpoints. Hence a line segment has one and only one midpoint.

Since $B$ lies between $A$ and $C$, the segment $AC$ is made up of two non-overlapping parts $AB$ and $BC$.

By Euclid's common notion 'the whole is equal to the sum of its parts', the whole segment $AC$ equals the sum of its parts $AB$ and $BC$: $AB + BC = AC.$

Hence proved.

The points lie in the order $A, B, C, D$ on a line.

Given: $AB = CD. \quad (1)$

Add $BC$ to both sides (Euclid's axiom: if equals are added to equals, the wholes are equal): $AB + BC = CD + BC. \quad (2)$

Now use the betweenness/whole-is-sum-of-parts property:

• $AB + BC = AC$ (since $B$ lies between $A$ and $C$),
• $CD + BC = BC + CD = BD$ (since $C$ lies between $B$ and $D$).

Substituting into (2): $AC = BD.$

Hence proved.

Statement: Two distinct lines cannot have more than one point in common.

Justification (by contradiction): Suppose two distinct lines $\ell$ and $m$ have two common points, say $P$ and $Q$.

Then both $\ell$ and $m$ pass through the two distinct points $P$ and $Q$. But Euclid's first postulate says that through two distinct points there passes exactly one line. This means $\ell$ and $m$ must be the same line — contradicting that they are distinct.

Hence two distinct lines can have at most one point in common.