Euclid's Elements Book I

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

Proposition I.15 · Theorem

If two straight lines cut one another, they make the vertical angles equal to one another.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let the straight lines AB, CD cut one another at the point E. The claim: the angle AEC is equal to the angle DEB, and the angle CEB to the angle AED. (All four ends can be dragged.)
  2. Since the straight line AE stands on the straight line CD, making the angles CEA, AED, these angles are equal to 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.
  3. Again, since the straight line DE stands on the straight line AB, making the angles AED, DEB, these angles are also equal to 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. Therefore the angles CEA, AED are equal to the angles AED, DEB. Let the angle AED be subtracted from each; then the remaining angle CEA is equal to the remaining angle BED.
    Postulate 4Common Notion 1Common Notion 3
    Postulate 4: That all right angles are equal to one another.
    Common Notion 1: Things which are equal to the same thing are also equal to one another.
    Common Notion 3: If equals be subtracted from equals, the remainders are equal.
  5. Similarly it can be proved that the angle CEB is equal to the angle DEA.