Page:First six books of the elements of Euclid 1847 Byrne.djvu/64

30 STRAIGHT line  falling on two parallel straight lines ( and ), makes the alternate angles equal to one another; and also the external equal to the internal and opposite angle on the same side; and the two internal angles on the same side together equal to two right angles.

For if the alternate angles and  be not equal, draw, making  =  (pr. 23). Therefore ||  (pr. 27.) and therefore two straight lines which intersect are parallel to the same straight line, which is impossible (ax. 12). Hence the alternate angles and  are not unequal, that is, they are equal:  =  (pr. 15);

∴ =, the external angle equal to the internal and opposite on the same side: if  be added to both, then  +  =  =  (pr. 13). That is to say, the two internal angles at the same side of the cutting line are equal to two right angles.

Q. E. D.