The first two books of the Elements of Euclid, with additional figures, notes, explanations, and deductions, by N. Pocock

therefore the angles CBE, EBD are equal to

the angles DBA, ABC;

but CBE, EBD are two right angles;

therefore DBA, ABC are together equal to two right angles. Wherefore, when a straight line, &c. Q. E. D.

PROP. XIV. THEOR.

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, these two straight lines shall be in one and the same straight line.

At the point B in the straight line
AB,

let the two straight lines BC, BD,
upon the opposite sides of AB,
make the adjacent angles ABC, ABD
equal together to two right angles.
BD is in the same straight line with CB.

For, if BD be not in the same straight line with CB, let BE be in the same straight line with it;

Therefore, because the straight line AB makes angles

with the straight line CBE, upon one side of it, the angles ABC, ABE are together

equal a to two right angles;

but the angles ABC, ABD are likewise together

equal to two right angles;

therefore the angles CBA, ABE are equal

to the angles CBA, ABD.

Take

away the common angle ABC,

the remaining angle ABE is equal to

the remaining angle ABD,

the less to the greater, which is impossible;

therefore BE is not in the same straight line with BC. And, in like manner, it may be demonstrated,

BOOK I.

a xiii. 1.

b 3 Ax.

BOOK L that no other can be in the same straight line with it but BD, which therefore is in the same straight line with CB. Wherefore, if at a point, &c. Q. E. D.

. xiii. 1.

b 3 Ax.

PROP. XV. THEOR.

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

Let the two straight lines AB, CD cut one another

in the point E;

the angle AEC shall be equal to the angle DEB,
and CEB to AED.

Because the straight

line AE

makes with CD the angles

CEA, AED,

these angles are together

equal a to two right angles.

Again, because the straight line DE makes with AB,
the angles AED, DEB,

these also are together equal a to two right angles;
and CEA, AED have been demonstrated to be equal
to two right angles;

wherefore the angles CEA, AED are equal to
the angles AED, DEB.

Take away the common angle AED,

and the remaining angle CEA is equal to
the remaining angle DEB.

In the same manner it can be demonstrated,
that the angles CEB, AED are equal.
Therefore, if two straight lines, &c. Q. E. D.

COR. 1. From this it is manifest, that,

If two straight lines cut one another, the angles they make at the point where they cut, are together equal to four right angles.

[For CEA and CEB are together equal to two right BOOK I. angles, and AED and DEB are together equal to two right angles, or all the four angles are equal to four right angles.]

COR. 2. And consequently that

All the angles made by any number of lines meeting in one point, are together equal to four right angles.

PROP. XVI. THEOR.

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

Let ABC be a triangle,

and let its side BC be produced to D,

the exterior angle ACD is greater than either of the interior

opposite angles CBA, BAC.

Bisecta AC in E,

join BE and produce it to F, and make EF equal to BE; join also FC.

Because AE is equal to EC, and BE to EF;

AE, EB are equal to CE, EF, each to each;

BOOK L and the angle AEB is equal to the angle CEF, because they are opposite vertical angles ;

b xv. 1.

therefore the base AB is equal to the base CF,
and the triangle AEB to the triangle CEF,

and the remaining angles to the remaining angles,
each to each, to which the equal sides are opposite;
wherefore the angle BAE is equal to the angle ECF;
but the angle ECD is greater than the angle ECF;
therefore the angle ACD is greater than BAE:

In the same manner, if the side BC be bisected, and AC pro-
duced to G, it may be demonstrated that the angle BCG,
which is equal to the angle ACD,

is greater than the angle ABC.

Therefore, if one side, &c. Q. E. D.

[Observe that the angle ACD has been proved greater than ABC, and that the angle BCG may be proved greater than BAC; and as ACD and BCG are equal, it follows that either of them is greater than either of the interior opposite angles ABC, BAC. The exterior angle may be greater than, equal to, or less than the interior angle which is adjacent to it, as may be seen in the annexed figures.]