Euclid's Elements Book I

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

Proposition I.29 · Theorem

A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side 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 EF fall on the parallel straight lines AB, CD. The claim: it makes the alternate angles AGH, GHD equal, the exterior angle EGB equal to the interior and opposite angle GHD, and the interior angles on the same side, BGH, GHD, equal to two right angles. (Try dragging G, H or B.)
  2. If the angle AGH is unequal to the angle GHD, let the angle AGH be greater. Let the angle BGH be added to each; then the angles AGH, BGH are greater than the angles BGH, GHD.
  3. But the angles AGH, BGH are equal to two right angles; therefore the angles BGH, GHD are less than two right angles.
    Proposition I.13
    Proposition I.13: 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.
  4. But straight lines produced indefinitely from angles less than two right angles meet; so AB, CD would meet, yet they are parallel and do not meet, which is impossible. Therefore the angle AGH is equal to the angle GHD.
    Postulate 5Definition 23
    Postulate 5: That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
    Definition 23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
  5. Again, the angle AGH is equal to its vertical angle EGB; therefore the angle EGB is also equal to the angle GHD.
    Proposition I.15Common Notion 1
    Proposition I.15: If two straight lines cut one another, they make the vertical angles equal to one another.
    Common Notion 1: Things which are equal to the same thing are also equal to one another.
  6. Let the angle BGH be added to each; then the angles EGB, BGH are equal to the angles BGH, GHD. The former are equal to two right angles; therefore the angles BGH, GHD are also equal to two right angles.
    Common Notion 2Proposition I.13
    Common Notion 2: If equals be added to equals, the wholes are equal.
    Proposition I.13: 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.