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RELFE BROTHERS' EUCLID SHEETS—Props. 1-26, Book I, are now published in a similar form to this.

PROPOSITION XXVI.

164.

If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz., either the sides adjacent to the equal angles in each, or the sides opposite to them; then shall the other sides be equal, each to each, and also the third angle of the one equal to the third angle of the other.

be two triangles which have the angles

Let

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equal to the angles each to each, namely, also one side equal to one side. First, let those sides be equal which are adjacent to the angles that are equal in the two triangles, namely, Then the other sides shall be equal, each to each, namely, and

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and the other angles to the other angles, each to each, to which the equal sides are

is, by the hypothesis, equal the less angle equal to the greater,

be not equal to
equal to

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and

to

the two sides,

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are equal to the two

one of them must be greater than the other. If possible, let

be greater

, because

each to each; and

therefore the base ;

is equal to the base

; but the angle

is equal to the angle

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Secondly, let the sides which are opposite to one of the equal angles in each triangle be equal to one another,

Then in this case likewise the other sides shall be equal,

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one of them must be greater than the other. If possible, let

be greater

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and join

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and

to

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and the angle

Then in the two triangles
to the angle

because

is

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; therefore the base

is equal to

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to which the equal sides are opposite;

is equal to the angle

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is equal to its interior and opposite angle

that is, is equal to

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; because is equal to

therefore the base

Wherefore, if two triangles, &c.

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and the other angles to the other angles, each to each,

; but the angle

that is, the ex

; which is impossible;

and

to

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,

is equal to the base

and the included

and the third

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

PROPOSITION XXV.

If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of one greater than the base of the other; the angle contained by the sides of the one which has the greater base, shall be greater than the angle contained by the sides, equal to them, of the other.

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to it, or less than it. If the angle

were equal to the angle

it must either be equal

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is

would be equal to the base

not equal to the angle

Again, if the angle

; but it is not equal, therefore the angle

were less than the angle

then the base

would

be less than the base

; but it is not less, therefore the angle

is not less than

the angle

; therefore the angle

; and it has been shewn, that the angle

Wherefore, if two triangles, &c.

is greater than the angle

is not equal to the angle

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

PROPOSITION XXIV.

If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle ntained by the two sides of one of them greater than the angle contained by the two sides equal to them, of e other; the base of that which has the greater angle, shall be greater than the base of the other.

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be two triangles, which have the two sides.

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each to each, namely,

equal to

and

to

equal to ; but the

greater than the angle

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each to each, and the angle

is equal to the base

is equal to the angle

And because

is equal to

in the triangle

equal to the angle

are equal to the two

; therefore the base

therefore the angle

is

s equal to the angle

; but the angle

is greater than the angle

herefore the angle

he angle

is also greater than the angle greater than the angle

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And because in the triangle

ind that the greater angle is subtended by the greater side; therefore the side
greater than the side
; but

han

Wherefore, if two triangles, &c.

the angle

is greater than the angle

was proved equal to

; therefore

is

is greater

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