Introduction to Euclid's Geometry: Axioms and Postulates Class 9: notes and important questions

Introduction to Euclid's Geometry: revision notes

1. Where geometry came from

Geometry began with practical work: marking out fields, building, and laying out altars. In India, the Sulbasutras (the Baudhayana Sulbasutra is the best known) give rules for making altars of exact shapes and sizes with pegs and cords, including squares, rectangles and the diagonal rule: the square on the diagonal of a rectangle equals the squares on its two sides together (the Baudhayana-Pythagoras theorem). In Egypt, surveyors re-marked fields and planned large buildings. Greek mathematicians asked why such rules are true and proved them from a few starting assumptions. Euclid, working in Alexandria around 300 BCE, collected this work in the Elements, thirteen books that build geometry step by step from definitions, axioms and postulates.

2. Definitions, axioms, postulates, theorems

• Definition: describes a term using simpler terms. Euclid described a point as that which has no part, a line as length without breadth, and a surface as having length and breadth only. Some terms (point, line, plane) cannot be defined without going round in circles, so we take them as undefined terms that we understand intuitively.
• Axiom: an assumption accepted without proof and used throughout mathematics (Euclid's "common notions").
• Postulate: an assumption accepted without proof that is specific to geometry.
• Theorem: a statement that is proved from definitions, axioms, postulates and earlier theorems.

3. Euclid's axioms (in our words)

1. Things 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 that coincide with one another are equal to one another.
5. The whole is greater than the part.
6. Things that are double the same thing are equal to one another.
7. Things that are halves of the same thing are equal to one another.

Modern editions of the Elements give five common notions, the first five above; the two about doubles and halves are found only in some later copies, and school courses keep them because they are useful.

Axioms of measurement. Every segment has a positive length. If \(C\) lies on segment \(AB\) between \(A\) and \(B\), then \(AC + CB = AB\). Angles add in the same way: if ray \(OC\) lies inside \(\angle AOB\), then \(\angle AOC + \angle COB = \angle AOB\).

4. Euclid's five postulates (in our words)

1. A straight line can be drawn from any point to any other point. (In fact exactly one: two distinct points fix a unique line.)
2. A straight line segment can be extended as far as we like in a straight line.
3. A circle can be drawn with any centre and any radius.
4. All right angles are equal to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles (\(180^\circ\)), the two lines, if extended, meet on that side.

Parallel lines are lines in a plane that never meet, however far they are extended. A statement equivalent to Postulate 5 (Playfair's axiom): through a point not on a given line, there is exactly one line parallel to the given line. Postulate 5 is far longer than the others, and for centuries mathematicians tried to prove it from the first four; it cannot be done.

5. Laying out a square with a given side

To lay out a square on a side \(AB\) with pegs and cords, as the altar builders had to: make a right angle at \(A\) and at \(B\), mark \(AD = BC = AB\) along them, and join \(CD\). The diagonal rule checks the work: in a square of side \(a\), each diagonal \(d\) satisfies \(d^2 = a^2 + a^2 = 2a^2\), so \(d = a\sqrt2\), and the two diagonals must be equal. A cord in the ratio \(3 : 4 : 5\) (or \(5 : 12 : 13\)) makes a right angle, because \(3^2 + 4^2 = 5^2\).

6. Justifying a construction from the axioms

Equilateral triangle on a segment \(AB\). Draw the circle with centre \(A\) through \(B\), and the circle with centre \(B\) through \(A\) (Postulate 3). Let them meet at \(C\); join \(CA\) and \(CB\) (Postulate 1). \(AC = AB\) (radii of the first circle) and \(BC = AB\) (radii of the second), so \(AC = BC\) (Axiom 1): all three sides are equal.

Every segment has exactly one mid-point. Suppose \(M\) and \(N\) were both mid-points of \(AB\). Then \(AM = \frac12 AB = AN\) (Axiom 7), so \(M\) and \(N\) are at the same distance from \(A\) along \(AB\): they coincide.

Worked example 1

\(A, B, C, D\) lie on a line in that order, with \(AB = CD\). Show that \(AC = BD\).

\(AC = AB + BC\) and \(BD = BC + CD\) (measurement). Since \(AB = CD\), adding \(BC\) to both gives \(AB + BC = CD + BC\) (Axiom 2), so \(AC = BD\).

Worked example 2

A transversal makes interior angles of \(84^\circ\) and \(91^\circ\) on the same side with two lines. Do the lines meet, and on which side?

\(84^\circ + 91^\circ = 175^\circ \lt 180^\circ\), so by Postulate 5 the lines meet on that side.

Common errors

• Mixing up axioms (for all of mathematics) and postulates (for geometry).
• Quoting the axiom number without saying what it states: write the axiom in words.
• Assuming a fact from the figure ("it looks equal") instead of giving a reason.
• Reading Postulate 5 the wrong way round: the lines meet on the side where the two interior angles add to less than \(180^\circ\).

Exam tips

• In a justification, give every step a reason: a definition, an axiom, a postulate or an earlier result.
• For a length question on one line, draw the points in order first, then use \(AC + CB = AB\).