Introduction to Euclid's Geometry: Axioms and Postulates

The One Thing

Euclidean geometry is a logical system built from precise definitions, accepted axioms and postulates, and step-by-step proofs. Euclid organized this system in the Elements, establishing a foundation for reasoning about points, lines, angles, figures, and their relationships.

Definitions and Results

• Euclid: An ancient Greek mathematician, often called the father of geometry, who organized geometry systematically in his work commonly known as the Elements.
• Geometry: The branch of mathematics that studies shapes, sizes, positions, lines, angles, and spaces.
• Definition: A precise explanation of the meaning of a mathematical term, such as point, line, or circle.
• Point: An exact location with no length, breadth, or thickness. A point has no dimensions.
• Line: A straight path extending endlessly in both directions and having length but no breadth.
• Line segment: A part of a line bounded by two endpoints. A finite line segment has a definite length.
• Ray: A part of a line that begins at one endpoint and extends endlessly in one direction.
• Axiom: A statement accepted as true without proof and generally used in many areas of mathematics.
• Postulate: A statement accepted as true without proof and used specifically as a foundation for geometry.
• Theorem: A mathematical statement proved using definitions, axioms, postulates, and earlier results.
• Proof: A logical sequence of statements establishing why a mathematical result is true. Each step must be justified by a definition, axiom, postulate, or previously established result.
• Incidence: The relationship between geometric objects, such as a point lying on a line.
• Collinear points: Two or more points that lie on the same straight line.
• Euclid’s five postulates:

1. A straight line can be drawn from any one point to any other point. 2. A terminated line can be produced indefinitely. 3. A circle can be drawn with any centre and any radius. 4. All right angles are equal to one another. 5. If a line intersects two lines and the interior angles on one side have sum less than two right angles, the two lines meet on that side when extended.

• Euclid’s fifth postulate: If a line falling on two lines makes the interior angles on the same side add to less than two right angles, the two lines, if extended indefinitely, meet on that side. It is related to the result that through a point not on a given line, exactly one parallel line can be drawn to the given line.
• Common axioms:

- Things equal to the same thing are equal to one another. - If equals are added to equals, the wholes are equal. - If equals are subtracted from equals, the remainders are equal. - Things that coincide with one another are equal. - The whole is greater than the part.

• Axioms and postulates: Both are starting assumptions and are not normally proved within the system in which they are used. In school geometry, axioms are generally mathematical truths applicable across areas, whereas postulates are geometry-specific assumptions.
• Geometric dimensions: A point has no dimensions; a line has length but no breadth; and a plane surface has length and breadth but no thickness.
• Two distinct lines: Two distinct lines cannot have more than one common point, because two different points determine exactly one straight line.

Worked Methods

Constructing a straight line between two points

• Begin with any two points.
• Apply the first postulate: a straight line can be drawn from any one point to any other point.
• The resulting line joins the two points.

Extending a terminated line

• Begin with a terminated line, or line segment.
• Apply the second postulate.
• Produce the line indefinitely beyond either endpoint.
• Distinguish the resulting infinite line from the original finite line segment, which has a definite length.

Drawing a circle

• Choose any centre.
• Choose any radius.
• Apply the third postulate.
• Draw a circle with the selected centre and radius.

Using equal right angles

• Identify the right angles involved.
• Apply the fourth postulate: all right angles are equal to one another.
• Treat the right angles as equal when establishing later geometric results.

Applying the fifth postulate

• Consider a line intersecting two other lines.
• Identify the interior angles on the same side of the intersecting line.
• Determine whether their sum is less than two right angles.
• If it is, extend the two lines indefinitely.
• Conclude that the two lines meet on that side.
• In its parallel-line interpretation, through a point not on a given line, exactly one parallel line can be drawn to the given line.

Developing a theorem by proof

• State the geometric result to be established.
• Identify the relevant definitions, axioms, postulates, and previously proved results.
• Use a diagram to clarify the geometric situation, without treating its visual appearance as proof.
• Proceed through a sequence of logically justified statements.
• Ensure that every statement follows from an accepted fact or an earlier result.
• Conclude the theorem once the required result has been established.

Establishing the relationship between two distinct lines

• Assume that two distinct lines have two common points.
• Use the fact that two different points determine exactly one straight line.
• Conclude that the two lines must in fact be the same line.
• Therefore, two distinct lines cannot have more than one common point.

Where It Goes Wrong

• Confusing a line segment with a line: a line segment has two endpoints and definite length, whereas a line extends endlessly in both directions.
• Treating a diagram’s appearance as proof: a diagram assists understanding, but the argument must depend on logical reasoning.
• Reversing the roles of axioms and postulates: axioms are generally applicable mathematical truths, while postulates are specifically used as foundations of geometry.
• Attempting to prove an axiom or postulate within the system: these statements are accepted as starting assumptions.
• Omitting the condition in Euclid’s fifth postulate: the interior angles on the same side must sum to less than two right angles, and the lines must be extended indefinitely.
• Failing to justify individual proof steps: every statement must follow from a definition, axiom, postulate, or previously established result.

What Gets Asked

• Define Euclid, geometry, point, line, line segment, ray, axiom, postulate, theorem, proof, incidence, and collinear points.
• State Euclid’s five postulates, including the precise condition in the fifth postulate.
• Distinguish between an axiom and a postulate.
• Explain the dimensions of a point, line, and plane surface.
• Distinguish between a finite line segment and an indefinitely extending line.
• State common axioms, including equality, addition and subtraction of equals, coincidence, and the whole being greater than the part.
• Explain why two distinct lines cannot have more than one common point.
• Describe how definitions, axioms, and postulates are used to establish a theorem.
• Explain the role and limitations of a geometric diagram in a proof.