Euclid's Elements Book I

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

Proposition I.16 · Theorem

In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let ABC be a triangle, and let its side BC be produced to D. The claim: the exterior angle ACD is greater than each of the interior and opposite angles CBA, BAC. (All three vertices can be dragged.)
    Postulate 2
    Postulate 2: To produce a finite straight line continuously in a straight line.
  2. Let AC be bisected at E.
    Proposition I.10
    Proposition I.10: To bisect a given finite straight line.
  3. Let BE be joined and produced to F, and let EF be made equal to BE.
    Postulate 1Postulate 2Proposition I.3
    Postulate 1: To draw a straight line from any point to any point.
    Postulate 2: To produce a finite straight line continuously in a straight line.
    Proposition I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
  4. Let FC be joined.
    Postulate 1
    Postulate 1: To draw a straight line from any point to any point.
  5. Since AE is equal to EC and BE to EF, and the vertical angle AEB is equal to the angle FEC,
    Proposition I.15
    Proposition I.15: If two straight lines cut one another, they make the vertical angles equal to one another.
  6. the base AB is equal to the base FC, the triangle ABE is equal to the triangle FEC, and the angle BAE is equal to the angle ECF.
    Proposition I.4
    Proposition I.4: If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively.
  7. But the angle ECD is greater than the angle ECF; therefore the angle ACD is greater than the angle BAE.
    Common Notion 5
    Common Notion 5: The whole is greater than the part.
  8. Similarly, if BC is bisected and AC produced to G, it can be proved that the angle BCG, which equals its vertical angle ACD, is greater than the angle ABC.
    Proposition I.15
    Proposition I.15: If two straight lines cut one another, they make the vertical angles equal to one another.