RELFE BROTHERS EUCLID SHEETS.

' .

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

PROPOSITION XVI.

If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

, and

Because is equal to

to ; the two sides

are equal to the two each to each, in the triangles

; and the angle is equal to the angle because they are opposite vertical angles; therefore the base is equal to the base and the triangle to the triangle

and the remaining angles of one triangle to the remaining angles of the other, each to each, to which the equal sides are opposite; wherefore the angle is equal to the angle

; but the angle is greater than the angle ; therefore the angle

Therefore, if one side of a triangle, &c.

RELFE BROTHERS' EUCLID SHEETS.

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

PROPOSITION XV.

If two straight lines cut one another, the vertical, or opposite angles shall be equal. Let the two straight lines

cut one another in the point Then the angle shall be equal to the angle

and the angle

to the

Because the straight line makes with at the point the adjacent angles

; these angles are together equal to two right angles. Again, because the straight line makes with at the point the adjacent angles

; these angles also are equal to two right angles; but the angles have been shewn to be equal to two right angles; wherefore the angles are equal to the angles

take
away

from each the common angle and the remaining angle is equal to the remaining angle

In the same manner it may be demonstrated, that the angle is equal to the angle

i

Therefore, if two straight lines cut one another, &c.

RELFE BROTHERS' EUCLID SHEETS,

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

PROPOSITION XIV.

If at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles ; then these two straight lines shall be in one and the same straight line.

At the point

upon the

in the straight line let the two straight lines
make the adjacent angles

together equal to two

opposite sides of

right angles.

Then

shall be in the same straight line with

if possible, let

be in the

For, if be not in the same straight line with same straight line with it.

; take

Then because meets the straight line ; therefore the adjacent angles are equal to two right angles; but the angles

are equal to two right angles; therefore the angles

are equal to the angles away from these equals the common angle

therefore the remaining angle is equal to the remaining angle ; the less angle equal to the greater, which is impossible: therefore is not in the same straight line with

And in the same manner it may be demonstrated, that no other can be in the same straight line with it but which therefore is in the same straight line with

Wherefore, if at a point, &c.