Euclid's Elements Book I

1. Let ABC be a triangle having the side AC greater than the side AB. The claim: the angle ABC is also greater than the angle BCA. (Drag the vertices, keeping AC longer than AB.)
2. From AC let AD be cut off equal to AB.

   Proposition I.3

   Proposition I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
3. Let BD be joined.

   Postulate 1

   Postulate 1: To draw a straight line from any point to any point.
4. Since the angle ADB is an exterior angle of the triangle BCD, it is greater than the interior and opposite angle DCB.

   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.
5. But the angle ADB is equal to the angle ABD, since the side AB is equal to AD; therefore the angle ABD is also greater than the angle ACB.

   Proposition I.5

   Proposition I.5: In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.
6. Therefore the angle ABC is much greater than the angle ACB.

   Common Notion 5

   Common Notion 5: The whole is greater than the part.

Being what it was required to prove.