RELFE BROTHERS' EUCLID SHEETS.

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.

at

For if the angle be equal to the angle each of them is a right angle. But if the angle be not equal to the angle

from the point draw right angles to Then the angles

are two right angles. And because the angle is equal to the angles

add the angle to each of these equals; therefore the angles

are equal to the three

angles
Again, because the angle is equal to the two angles

add to each of these equals the angle ; therefore the angles

are equal to the three 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

; but the angles two right angles; therefore the angles

are together equal to two right angles. Wherefore, when a straight line, &c.

are

RELFE BROTHERS' EUCLID SHEETS.

'

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

be the given straight line, which may be produced any length both ways,

and let

be a point without it.

It is required to draw a straight line perpendicular to

from the point

:

describe the circle

meeting produced if necessary, in and bisect in

and join

Then the straight line drawn from the given point shall be perpendicular to the straight line Join and

Because is equal to and is common to the triangles the two sides are equal to the two

each to each ; and the base

is equal to the base

; therefore the angle

is equal to the angle

; and

these are adjacent angles.

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 has been drawn to the given

straight line