Euclid, the Greek mathematician, laid the foundations of geometry around 300 BCE in his work "Elements." He started with a small set of assumptions and derived all other results logically.
- Definitions (selected):
- A point has no part.
- A line is breadthless length.
- A straight line lies evenly with the points on itself.
- A surface has length and breadth only.
- 1.Euclid's Postulates:
- 2.A straight line may be drawn from any one point to any other point.
- 3.A terminated line can be produced indefinitely.
- 4.A circle can be drawn with any centre and any radius.
- 5.All right angles are equal to one another.
- 6.Parallel Postulate: If a straight line falls on two straight lines making interior angles on the same side less than two right angles, then the two lines, if produced indefinitely, meet on that side.
- 1.Euclid's Axioms (common notions):
- 2.Things equal to the same thing are equal to each other.
- 3.If equals are added to equals, the wholes are equal.
- 4.If equals are subtracted from equals, the remainders are equal.
- 5.Things that coincide are equal.
- 6.The whole is greater than the part.
Theorem: Two distinct lines cannot have more than one point in common.
Proof: Suppose two lines l and m meet at two points P and Q. But postulate 1 says only one line can pass through two points — contradiction. Hence, two lines meet in at most one point.
If AC = BD, prove that AB = CD given A, B, C, D are collinear in order A-B-C-D.
AC = AB + BC and BD = BC + CD.
Since AC = BD, AB + BC = BC + CD.
By Axiom 3 (subtracting BC): AB = CD.
State the postulate that justifies drawing a circle with centre O and radius 5 cm.
Postulate 3: A circle can be drawn with any centre and any radius.
Which axiom is used when we say "If x = y and y = z, then x = z"?
Axiom 1: Things equal to the same thing are equal to each other.
Two angles ABC and PQR each measure 90 degrees. By which postulate are they equal?
Postulate 4: All right angles are equal to one another.
Name the postulate that allows a line segment to be extended to a full line.
Postulate 2: A terminated line can be produced indefinitely.
If AB = 5 cm and BC = 5 cm and B is between A and C, what is AC?
AC = AB + BC = 5 + 5 = 10 cm (by Axiom 2).
Is it possible for three lines to intersect at the same point?
Yes — three lines can be concurrent, meaning they all pass through one common point. This is not ruled out by Euclid's postulates.
Common mistakes
Students confuse axioms with postulates. Axioms are universal truths in all of mathematics; Euclid's postulates are specific geometric assumptions. Also, Euclid's fifth postulate is equivalent to the parallel postulate and cannot be proved from the others — different assumptions give non-Euclidean geometries.
Summary
Euclid's approach uses undefined terms, definitions, postulates and axioms to build geometry logically. Five postulates form the basis of Euclidean geometry. Results proved from these are called theorems.