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← Introduction to Euclid's Geometry: Axioms and Postulates

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Lesson 10 of 11

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Eight questions, one for each idea.

Eight ideas geometry needs before it can prove anything
IdeaThe rule
ExamplesA pile of passes is not a reason. One failing case ends the claim.
DefinitionsUse only words already defined. Point and line are undefined.
The two axiomsThings equal to the same thing are equal. Equals added to equals stay equal.
Axiom or postulateA statement about rice is an axiom. A statement about a point, line, circle or angle is a postulate.
Postulates 1 to 3Join any two points. Extend any line. Draw any circle.
ConstructionEach drawing step is allowed by a postulate, and needs what it uses to exist first.
The equal squareIts side, times itself, gives the area. Equal ground does not mean an equal way round.
The fifth postulateAngles on one side adding to under 180° mean the lines meet on that side.

A claim holds for n = 0 to 39. Then n = 40 gives 1681 = 41 × 41. What do the 40 passes prove?

  • The claim is true for every n, since it passed forty cases
  • Nothing, since a pile of passes is examples, not a reason
  • The claim fails only at 41, the next number after the passes

A card says "Parallel lines keep the same direction." Point, line and ray are defined. What is wrong?

  • "Direction" has no definition yet
  • Nothing. It sounds right, so it can stay
  • It defines the word with itself

Sack A balances crate M. Sack B balances crate M. Sack C balances nothing. What follows about C?

  • A, B and C are all equal to one another
  • Nothing. C has no link to A or B
  • C is heavier than A and B

"All right angles are equal to one another." Axiom or postulate?

  • Axiom, because it says "equal"
  • Neither, since it is only a definition
  • Postulate, because it names an angle

Which postulate draws a circle of any centre and radius?

  • The third
  • The first
  • The second

You mark the crossings before drawing circle B. What happens?

  • It is refused. Both circles must exist first
  • It works. One circle is enough to mark a crossing
  • It works if the radius is small

A 16-by-1 corridor and a 4-by-4 square cover 16. Their ways round?

  • They are equal, as both cover 16
  • Both are 16, the same as the ground
  • They differ: 34 and 16

Two lines cross a third. The angles on one side add to exactly 180°. What does the fifth postulate say?

  • They meet on that side
  • Nothing
  • They never meet

This chapter is Introduction to Euclid's Geometry: Axioms and Postulates. Open your book at this chapter. Use these ideas as you read.

Next: Lines and Angles. It proves the first results from this chapter's agreements.

Reveal every answer to finish this lesson.