In any triangle the greater angle is subtended by the greater side.
We will start with a triangle ABC in which angle ABC is greater than angle ACB.
We will now prove that side AC is greater than side AB.
Suppose that this was not true. In other words, AC is not greater than side AB. Then AC would either be equal to AB or less than AB.
AC equal to AB?
According to Proposition 5 (the two angles at the base of an equilateral triangle are equal) AC cannot be equal to AB, because they would be two equal sides in an isosceles triangle and that would imply that angles ABC and ACB would be equal which they are not.
AC less than AB?
AC cannot be less than AB, for that would violate Proposition 18 (in any triangle the greater side subtends the greater angle) in that the greater side (AB) does not subtend the greater angle (ABC).
Therefore AC is greater than AB.
Q.E.D.
Questions? Comments? Please leave feedback – George