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PROPOSITIONS 1-26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS.
The angles which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles.
Let the straight line
make with , upon one side of it, the angles
Then these shall be either two right angles, or, shall be together, equal to two right angles.
But the angles
have been proved equal to the same three angles; and
things which are equal to the same thing are equal to one another; therefore the angles
are equal to the angles
two right angles; therefore the angles
; but the angles
are together equal to two right angles.
Wherefore, when a straight line, &c.
PROPOSITIONS 1–26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS.
To draw a straight line perpendicular to a given straight line of unlimited length, from a given point without it.
be the given straight line, which may be produced any length both ways, be a point without it.
It is required to draw a straight line perpendicular to
from the point
But when a straight line standing on another straight line, makes the adjacent angles equal to one another, each of them is a right angle, and the straight line which stands upon the other is called a perpendicular to it.
Therefore from the given point a perpendicular straight line
has been drawn to the given