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Year 9 Maths — Euclid's geometry

Tell an axiom from a postulate, and use Euclid's first assumptions to justify a step.

Name: ________________________

Before you start

Four drawings, and the only difference between them is where they stop. Find out what each one is called.

1234
  1. 1. A point — no length at all
  2. 2. A line segment — an end at each side
  3. 3. A ray — one end, and no end the other way
  4. 4. A straight line — no ends at all

A point has no length, a segment has two ends, a ray has one and a line has none. Euclid's opening definitions are this list and very little else — everything afterwards is built out of these four.

  1. 1.How many straight lines can be drawn through a single point?

    • a) Infinitely many
    • b) Exactly two
    • c) None
    • d) Exactly one

  2. 2.Sort each statement by whether Euclid assumed it or proved it.

    Groups: Assumed without proof · Proved from the others

    • The base angles of an isosceles triangle are equal
    • The whole is greater than the part
    • All right angles are equal to one another
    • A straight line may be drawn from any point to any other point
    • Two distinct lines cannot have more than one point in common
    • The angles of a triangle add up to 180°

  3. 3.If equals are subtracted from equals, the remainders are equal. Taking 8 from both sides of y + 8 = 15 gives what value of y?

  4. 4.Euclid set all of this out in thirteen books, collected under one title. What is that title? One word.

  5. 5."Two distinct straight lines cannot have more than one point in common." In Euclid's scheme this is...

    • a) one of the axioms
    • b) one of the five postulates
    • c) a definition
    • d) a theorem, proved from the postulates

  6. 6.Euclid's axiom says the whole is greater than the part. A segment 12 units long is cut into two parts and one of them is 5 units. How long is the other, in units?

  7. 7.In Euclid's system, what is a postulate?

    • a) A word being given a meaning
    • b) A statement proved from earlier ones
    • c) An assumption made about geometry in particular
    • d) A measurement taken off a drawing

  8. 8.If a = b and b = c, which of Euclid's axioms lets you write a = c?

    • a) Things which are equal to the same thing are equal to one another
    • b) If equals are added to equals, the wholes are equal
    • c) The whole is greater than the part
    • d) Things which coincide with one another are equal

Answer key — Year 9 Maths — Euclid's geometry

  1. 1. a) Infinitely many
  2. 2. A straight line may be drawn from any point to any other point → Assumed without proof; All right angles are equal to one another → Assumed without proof; The whole is greater than the part → Assumed without proof; The angles of a triangle add up to 180° → Proved from the others; Two distinct lines cannot have more than one point in common → Proved from the others; The base angles of an isosceles triangle are equal → Proved from the others
  3. 3. 7
  4. 4.
    • Elements
    • The Elements
  5. 5. d) a theorem, proved from the postulates
  6. 6. 7
  7. 7. c) An assumption made about geometry in particular
  8. 8. a) Things which are equal to the same thing are equal to one another