RELFE BROTHERS' EUCLID SHEETS.

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

PROPOSITION VIII.

If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal; the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides equal to them, of the other.

be applied to

then because

wherefore

shall be equal to the angle

For, if the triangle

is equal to

so that the point
be on
and the straight line
therefore the point

on

;

shall coincide with the point

;

coinciding with

and

shall coincide with

; for,

same base, and upon the same side of it, there can be two triangles which have their sides which are terminated in one extremity of the base, equal to one another, and likewise those sides which are terminated in the other extremity; but this is impossible.

Therefore, if two triangles have two sides, &c.

RELFE BROTHERS' EUCLID SHEETS.

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

PROPOSITION VII.

Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base, equal to one another, and likewise those which are terminated in the other extremity.

and upon the same side of it, let there be two triangles

Join

First. When the vertex of each of the triangles is without the other triangle.

Because

is equal to

angle

; but the angle

is greater than

the side

is equal to

; but the angle

both equal to, and greater than the angle

in the triangle
is greater than the angle
; much more therefore is the angle
in the triangle
therefore the angle
was proved greater than the angle
; which is impossible.

therefore the angle

greater than

; therefore also the angle

is equal to the angle hence the angle

is

to and

therefore the angles

to one another; but the angle is greater than the angle Again, because

to

and join

Then because

is equal to

upon the other side of the base

in the are equal

is greater than the angle ; much more then is the angle in the triangle

is both equal to, and greater than the angle

; which is impossible.

Thirdly. The case in which the vertex of one triangle is upon a side of the other, needs no demonstration.

Therefore, upon the same base and on the same side of it, &c.