*EUCLID'S ELEMENTS*.

PROPOSITION 18. *THEOREM*.

The greater side of every triangle has the greater angle opposite to it.

Let *ABC* be a triangle, of which the side *AC* is greater than the side *AB*: the angle *ABC* is also greater than the angle *ACB*.

Because *AC* is greater than *AB*, make *AD* equal to *AB*, [I. 3.

and join *BD*.

Then, because *ADB* is the exterior angle of the triangle *BDC*, it is greater than the interior opposite angle *DCB*. [I. 16.

But the angle *ADB* is equal to the angle *ABD*, [I. 5.

because the side *AD* is equal to the side *AB*. [*Constr*.

Therefore the angle *ABD* is also greater than the angle *ACB*.

Much more then is the angle *ABC* greater than the angle *ACB*. [*Axiom* 9.

Wherefore, *the greater side* &c. q.e.d.

PROPOSITION 19. *THEOREM*.

The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it.

Let *ABC* be a triangle, of which the angle *ABC* is greater than the angle *ACB*: the side *AC* is also greater than the side *AB*.

For if not, *AC* must be either equal to *AB* or less than *AB*.

But *AC* is not equal to *AB*,

for then the angle *ABC* would be equal to the angle *ACB*; [I. 5.

but it is not; [*Hypothesis*.

therefore *AC* is not equal to *AB*. Neither is *AC* less than *AB*,

for then the angle *ABC* would be less than the angle *ACB*; [I. 18.

but it is not; [*Hypothesis*.