Euclid's Elements Book I

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

Proposition I.17 · Theorem

In any triangle two angles taken together in any manner are less than two right angles.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let ABC be a triangle. The claim: any two angles of the triangle ABC are together less than two right angles. (Try dragging the vertices.)
  2. Let BC be produced to D.
    Postulate 2
    Postulate 2: To produce a finite straight line continuously in a straight line.
  3. Since the angle ACD is an exterior angle of the triangle ABC, it is greater than the interior and opposite angle ABC.
    Proposition I.16
    Proposition I.16: In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.
  4. Let the angle ACB be added to each; then the angles ACD, ACB are greater than the angles ABC, BCA.
  5. But the angles ACD, ACB are equal to two right angles; therefore the angles ABC, BCA 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.
  6. Similarly it can be proved that the angles BAC, ACB, and also the angles CAB, ABC, are less than two right angles.