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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.

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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.

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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

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;

to the angle

and the angle

to the angle

And because the whole

proved to be equal to
each to each; and the angle

; 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

has been proved to be equal to the angle

also the base

is common to the two triangles
remaining angles, each to each, to which the equal sides are opposite; therefore the angle

; wherefore these triangles are equal, and their

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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.

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

to the

triangle

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

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shall fall on

because the angle

also the point
point was shewn to coincide with the point
with the base ; because the point

shall coincide with the point

do not coincide with the base enclose a space, which is impossible.

Therefore the base

triangle

is equal to the angle because

the straight line

therefore

is equal to

; but the shall coincide

if the base

and

would

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does coincide with coincides with the whole triangle remaining angles of one triangle coincide with the equal to them, viz., the angle

to the angle

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and is equal to it; also the

remaining angles of the other, and are

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Therefore, if two triangles have two sides of the one equal to two sides, &c.

to

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