In any triangle the side opposite the greater angle is greater.

Let *ABC* be a triangle having the angle *ABC* greater than the angle *BCA.*

I say that the side *AC* is greater than the side *AB.*

If not, either *AC* equals *AB* or it is less than it.

Now *AC* does not equal *AB,* for then the angle *ABC* would equal the angle *ACB,* but it does not. Therefore *AC* does not equal *AB.*

Neither is *AC* less than *AB,* for then the angle *ABC* would be less than the angle *ACB,* but it is not. Therefore *AC* is not less than *AB.*

And it was proved that it is not equal either. Therefore *AC* is greater than *AB.*

Therefore *in any triangle the side opposite the greater angle is greater.*

Q.E.D.

Without going into details, the law of sines contains more precise information about the relation between angles and sides of a triangle than this and the last proposition did. The law of sines states that
where By means of the law of sines the size of a angle can be related directly to the length of the opposite side. |