If two straight lines cut one another, they make the vertical angles equal to one another.
For let the straight lines AB,
CD cut one another
at the point E;
I say that the angle AEC is equal
to the angle DEB,
and the angle CEB
to the angle AED.
For, since the straight line AE stands
on the straight line CD,
making the angles CEA,
AED,
the angles CEA,
AED are equal to
two right angles.
[I. 13]
Again, since the straight line DE
stands on the straight line
AB, making the
angles AED,
DEB,
the angles AED,
DEB are equal to
two right angles
[I. 13]
But the angles CEA,
AED were also proved equal
to two right angles;
therefore the angles CEA,
AED are equal to the
angles AED,
DEB.
[Post. 4 and C.N. 1]
Let the angle AED be subtracted
from each;
therefore the remaining angle CEA
is equal to the remaining
angle BED.
[C.N. 3]
Similarly it can be proved that the angles CEB, DEA are also equal.
Therefore, etc. Q.E.D.
[PORISM. From this it is manifest that, if two straight lines cut one another, they will make the angles at the point of section equal to four right angles.]
Book I: Euclid, Elements, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)
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