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

PROPOSITION XVI.

If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite

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because they are opposite vertical angles; therefore the base

is equal to the angle

is equal to the base

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remaining angles of one triangle to the remaining angles of the other, each to each, to

which the equal sides are opposite; wherefore the angle

; but the angle

is greater than the angle

is equal to the angle ; therefore the angle

is greater than the angle

In the same manner, if the side may be demonstrated that the angle angle

Therefore, if one side of a triangle, &c.

be bisected, and

be produced to

; it

that is, the angle

is greater than the

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

PROPOSITION XV.

If two straight lines cut one another, the vertical, or opposite angles shall be equal.

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at the point , the adjacent

; these angles are together equal to two right angles.

Again, because the straight line

angles

makes with

; these angles also are equal to two right angles; but the angles have been shewn to be equal to two right angles; wherefore the angles ; take away from each the common

are equal to the angles

is equal to the remaining angle

angle

and the remaining angle the same manner it may be demonstrated, that the angle

In

is equal to the angle

Therefore, if two straight lines cut one another, &c.

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

PROPOSITION XIV.

If at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles; then these two straight lines shall be in one and the same straight line.

At the point opposite sides of

in the straight line let the two straight lines
make the adjacent angles

upon the

together equal to two

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therefore the adjacent angles are equal to

are equal to two right angles; but the angles two right angles; therefore the angles

; take

away from these equals the common angle

angle

is equal to the remaining angle

which is impossible: therefore

are equal to the angles

therefore the remaining

; the less angle equal to the greater,

is not in the same straight line with

And in the same manner it may be demonstrated, that no other can be in the same straight line with it but which therefore is in the same straight line with

Wherefore, if at a point, &c.

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