Euclid's Elements Book I

23 definitions · 5 postulates · 5 common notions · 48 propositions

Proposition I.13 · Theorem

If a straight line set up on a straight line makes angles, it will make either two right angles or angles equal to two right angles.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let the straight line AB set up on the straight line CD make the angles CBA, ABD. The claim: the angles CBA, ABD are either two right angles or equal to two right angles. (Try dragging A or B.)
  2. If the angle CBA is equal to the angle ABD, they are two right angles.
    Definition 10
    Definition 10: When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
  3. If not, let BE be drawn from the point B at right angles to CD; therefore the angles CBE, EBD are two right angles.
    Proposition I.11
    Proposition I.11: To draw a straight line at right angles to a given straight line from a given point on it.
  4. Since the angle CBE is equal to the two angles CBA, ABE, let the angle EBD be added to each; therefore the angles CBE, EBD are equal to the three angles CBA, ABE, EBD.
    Common Notion 2
    Common Notion 2: If equals be added to equals, the wholes are equal.
  5. Again, since the angle DBA is equal to the two angles DBE, EBA, let the angle ABC be added to each; therefore the angles DBA, ABC are equal to the three angles DBE, EBA, ABC.
    Common Notion 2
    Common Notion 2: If equals be added to equals, the wholes are equal.
  6. So the angles CBE, EBD and the angles DBA, ABC are equal to the same three angles, and therefore to one another. The angles CBE, EBD are two right angles; therefore the angles DBA, ABC are also equal to two right angles.
    Common Notion 1
    Common Notion 1: Things which are equal to the same thing are also equal to one another.