Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/42

 PROPOSITION 13. THEOREM. The angles which one straight line makes with another straight line on one side of it, either are two right angles, or are together equal to two right angles. Let the straight line AB make with the straight line CD, on one side of it, the angles CBA, ABD: these either are two right angles, or are together equal to two right angles. For if the angle CBA is equal to the angle ABD, each of them is a right angle. But if not, from the point B draw BE at right angles to CD; therefore the angles CBE, EBD are two right angles, Now the angle CBE is equal to the two angles CBA, ABE; to each of these equals add the angle EBD; therefore the angles CBE, EBD are equal to the three angles CBA, ABE, EBD. Again, the angle DBA is equal to the two angles DBE, EBA; to each of these equals add the angle ABC; therefore the angles DBA, ABC are equal to the three angles DBE, EBA, ABC. But the angles CBE, EBD have been shewn to be equal to the same three angles. Therefore the angles CBE, EBD are equal to the angles DBA, ABC But CBE, EBD are two right angles; therefore DBA, ABC are together equal to two right angles. Wherefore, the angles &c.