Page:First six books of the elements of Euclid 1847 Byrne.djvu/84

F the ſquare of one ſide () of a triangle is equal to the ſquares of the other two ſides ( and ), the angle ( ) ſubtended by that ſide is a right angle.

Draw $\perp$ and $=$ (prs.11.3.)

and draw alſo.

Since $=$ (conſt.)

$^{2}=$ $^{2}$;

$\therefore$ $^{2}+$ $^{2}=$ $^{2}+$ $^{2}$,

but $^{2}+$ $^{2}=$ $^{2}$ (pr. 47.),

and $^{2}+$ $^{2}=$ $^{2}$ (hyp.)

$\therefore$ $^{2}=$ $^{2}$,

$\therefore$ $=$ ;

and $\therefore$ $=$ (pr. 8.),

conſequently is a right angle.

Q. E. D.