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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 ikewise 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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shall be equal to the angle

so that the point be on and the straight line

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For, if the triangle

on

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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.

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

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

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

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

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Join

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

is equal to the

therefore the angle
; therefore also the angle

Because

is equal to

in the triangle

angle

; but the angle

is greater than

the side

is equal to

; but the angle

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

greater than

both equal to, and greater than the angle

Secondly. Let the vertex

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Again, because

is equal to the angle

hence the angle

is

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

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; much more then is the angle
in the triangle

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; therefore also the angle
greater than the angle

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is both equal to, and greater than the angle

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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