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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 kewise their bases equal; the angle which is contained by the two sides of the one shall be equal to the ngle contained by the two sides equal to them, of the other.

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be on

Then the angle shall be equal to the angle

For, if the triangle e applied to so that the point

and the straight line on

; hen because is equal to therefore the point shall coincide with the point ; therefore coinciding with and shall coincide with

; for, the base coincide with the base but the sides

, do not coincide

rith the sides
but have a different situation as

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

Therefore, if the base coincide with the base the sides

cannot

ut coincide with the sides

; wherefore likewise the angle

coincides

rith the angle

and is equal to it.

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.

If it be possible, on the same base

which have their sides to one another, and likewise their sides

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

of the base, equal that are terminated in

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is greater than

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

in the triangle

therefore the angle is equal to the angle ; but the angle is greater than the angle ; therefore also the angle ; much more therefore is the angle

Again, because the side is equal to

in the triangle

therefore the angle is equal to the angle ; but the angle

is was proved greater than the angle hence the angle both equal to, and greater than the angle i ; which is impossible. Secondly. Let the vertex of the triangle

fall within the triangle

greater than

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and to

Then because is equal to in the triangle therefore the angles

upon the other side of the base

, are equal to one another; but the angle is greater than the angle ; therefore also the angle is greater than the angle ; much more then is the angle greater than the angle Again, because is equal to in the triangle

therefore the angle is equal to the angle but the angle has been proved greater than

wherefore the angle 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.

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