PROPOSITIONS 1-26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS.

PROPOSITION XIII.

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

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.

PROPOSITION XII.

To draw a straight line perpendicular to a given straight line of unlimited length, from a given point without it.

Let

and let

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