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.
|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 a denotes the side opposite angle A, b denotes the side opposite angle B, and c denotes the side opposite angle C. In other words, the sine of an angle in a triangle is proportional to the opposite side.
By means of the law of sines the size of a angle can be related directly to the length of the opposite side.