Book 1
17
Similarly we can prove that the angles BAC, ACB are also less than two right angles, and so are the angles CAB, ABC as well.
17
Therefore etc.
17
Q. E. D.
Proposition 18.
18
Enunciation In any triangle the greater side subtends the greater angle.
18
Proof. For let ABC be a triangle having the side AC greater than AB;
18
I say that the angle ABC is also greater than the angle BCA.
18
For, since AC is greater than AB, let AD be made equal to AB [I. 3], and let BD bejoined.
18
Then, since the angle ADB is an exterior angle of the triangle BCD,
18
it is greater than the interior and opposite angle DCB. [I. 16]
18
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; therefore the angle ABC is much greater than the angle ACB.
18
Therefore etc.
18
Q. E. D.
Proposition 19.
19
Enunciation In any triangle the greater angle is subtended by the greater side.
19
Proof. Let ABC be a triangle having the angle ABC greater than the angle BCA;
19
I say that the side AC is also greater than the side AB.
19
For, if not, AC is either equal to AB or less.
19
Now AC is not equal to AB; for then the angle ABC would also have been equal to the angle ACB; [I. 5] but it is not; therefore AC is not equal to AB.
19
Neither is AC less than AB, for then the angle ABC would also have been less than the angle ACB; [I. 18] but it is not; therefore AC is not less than AB.
