RELFE BROTHERS' EUCLID SHEETS.

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

PROPOSITION VI.

If two angles of a triangle be equal to each other; the sides also which subtend, or are opposite to, the equal angles, shall be equal to one another.

RELFE BROTHERS' EUCLID SHEETS.

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

PROPOSITION V.

The angles at the base of an isosceles triangle are equal to each other; and if the equal sides be produced, the angles on the other side of the base shall be equal.

sides

is equal to
shall be equal to the angle

Let

be an isosceles triangle of which the side
be produced to and Then the angle

and let the equal

the greater, cut

are equal to the two

also the remaining angles of the one are equal to the remaining angles of the other, each to each, to

which the equal sides are opposite; viz., the angle

to the angle

and the angle

to the angle

And because the whole

; hence, because the two sides

is equal to the whole
is equal to the remainder
; and
are equal to the two

,

of which the parts

has been

are equal; therefore the remainder proved to be equal to

each to each; and the angle
is common to the two triangles

; wherefore these triangles are equal, and their

remaining angles, each to each, to which the equal sides are opposite; therefore the angle

And, since it has been demonstrated, that the whole angle

is

the

parts of which, the angles

equal to the remaining angle

is equal to the whole are also equal; therefore the remaining angle which are the angles at the base of the triangle

is

; and it

has also been proved, that the angle

is equal to the angle

which are the angles upon

the other side of the base. Therefore the angles at the base, &c.

RELFE BROTHERS' EUCLID SHEETS.

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

PROPOSITION IV.

If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by those sides equal to each other; they shall likewise have their bases or third sides equal, and the two triangles shall be equal, and their other angles shall be equal, each to each, viz., those to which the equal sides are opposite.

be two triangles, which have the two sides each to each, viz.,

to

and

to

equal to the and the included

Let two sides

angle

equal to the included angle

Then shall the base

be equal to the base

; and the triangle

triangle

; and the other angles to which the equal sides are opposite shall be equal,

each to each, viz., the angle

to the angle

and the angle

to the

to the

shall coincide with the point, because

was shewn to coincide with the point ; wherefore the base

shall coincide

; then the point

may

the straight line

therefore

is equal to

; but the

coinciding with and with

the two straight lines

and

would

and is equal to it; and the whole

and is equal to it; also the

e triangle coincide with the remaining angles of the other, and are

and the angle

les have two sides of the one equal to two sides, &c.

to