ar

I

it, t

or a

Book I.

let BE be in the same straight line with it; therefore, be-
cause the straight line AB makes angles with the straight
line CBE, upon one side of it, the angles ABC, ABE are
together equal to two right angles: but the angles ABC, a 13. 1.
ABD are likewise together equal to two right angles; there-
fore the angles CBA, ABE are equal to the angles CBA,
ABD: take away the common angle ABC, and the remaining
angle ABE is equal to the remaining angle ABD, the less b 3. Ax.
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, 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.

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.

In b 3. Ax.

are also together equal to two right angles; therefore the
two angles CEA, AED are equal to the two AED, DEB.
Take away the common angle AED, and the remaining
angle CEA is equal to the remaining angle DEB.
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 which they make at the point of
their intersection are together equal to four right angles.

COR. 2. And hence, all the angles made by any number of straight lines, meeting in one point, are together equal to four right angles.

Monely

PROP. XVI. THEOR.

IF one side of a triangle be produced, the exterior angle is greater than either of the interior and 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, and produce AC to G.

b

Because AE is equal to EC,
and BE to EF, AE, EB are
equal to CE, EF, each to B
each; and the angle AEB is
equal to the-angle CEF,
because they are vertical an-
gles; therefore the base AB
is equal to the base CF,
and the triangle AEB to the

C

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 ECD, that is, ACD, is greater than BAE. In the same manner, if the side BC Be bisected, it may be demonstrated that the angle BCG, that is the angle ACD, is greater than the angle ABC. Therefore, if one side, &c. Q. E. D.

PROP. XVII. THEOR.

ANY two angles of a triangle are together less than two right angles,

Let ABC be any triangle; any two of its angles together are less than two right angles.

B

C

D

a 16. 1.

Produce BC to D; and because ACD is the exterior angle of the triangle ABC, ACD is greater a than the interior and opposite angle ABC; to each of these add the angle ACB; therefore the angles ACD, ACB are greater than the angles ABC, ACB; but ACD, ACB are together equal to two right b 13. 1. angles; therefore the angles ABC, BCA are less than two right angles. In like manner, it may be demonstrated, that BAC, ACB, as also CAB, ABC, are less than two right angles. Therefore, any two angles, &c. Q. E. D.

PROP. XVIII. THEOR.

THE greater side of every triangle has the greater angle opposite to it.

Let ABC be a triangle, of which the side AC is greater than the side AB; the angle ABC is also greater than the angle BCA.

B

A

D

Ca 3.1.

From AC, which is greater than AB, cut offa AD equal to AB, and join BD; and because ADB is the exterior angle of the triangle BDC, it is greater than the interior and opposite b 16. 1. angle DCB; but ADB is equal to ABD, because the side AB is equal to the side AD; therefore the angle ABD is likewise greater than the angle ACB; wherefore much more is the angle ABC greater than ACB. Therefore the greater side, &c. Q. E. Ď.

C 5. 1.

Book I.

a 5. 1.

b 18. 1.

a 3. 1.

5. 1.

PROP. XIX. THEOR.

THE greater angle of every triangle is subtended by the greater side, or has the greater side opposite

to it.

Let ABC be a triangle, of which the angle ABC is greater than the angle BCA; the side AC is likewise greater than the side AB.

For, if it be not greater, AC must either be equal to AB or less than it; it is not equal, because then the angle ABC would be equal to the angle ACB; but it is not; therefore AC is not equal to AB; neither is it less; because then the angle ABC would be less than the angle ACB; but it is not; therefore the side AC is not less than AB; and it has been shown that it is not equal to AB; therefore AC is greater than AB. Wherefore the greater angle, &c. Q. E. D.

B

PROP. XX. THEOR.

ANY two sides of a triangle are together greater than the third side.

Let ABC be a triangle; any two sides of it together are greater than the third side, viz. the sides BA, AC greater than the side BC; and AB, BC greater than AC; and BC, CA greater than AB.

Produce BA to the point
D, and make a AD equal to
AC; and join DC.

Because DA is equal to
AC, the angle ADC is like-
wise equal to ACD; but
the angle BCD is greater

than the angle ACD; there

fore the angle BCD is greater than the angle ADC; and be

Book I.

cause the angle BCD of the triangle DCB is greater than its angle BDC, and that the greatere side is opposite to the greater angle: therefore the side DB is greater than the side c 19. 1. BC; but DB is equal to BA and AC together; therefore BA and AC together are greater than BC. In the same manner it may be demonstrated, that the sides AB, BC are greater than CA, and BC, CA greater than AB. Therefore any two sides, &c. Q. E. D.

PROP. XXI. THEOR.

IF from the ends of one side of a triangle there N. be drawn two straight lines to a point within the triangle, these two lines shall be less than the other two sides of the triangle, but shall contain a greater angle.

Let the two straight lines BD, CD be drawn from B, C, the ends of the side BC of the triangle ABC, to the point D within it; BD and DC are less than the other two sides BA, AC of the triangle, but contain an angle BDC greater than the angle BAC.

A

E

Produce BD to E; and because two sides of a trianglea are a 20. 1. greater than the third side, the two sides BA, AE of the triangle ABE are greater than BE. To each of these add EC; therefore the sides BA, AC are greater than BE, EC. Again, because the two sides CE, ED, of the triangle CED, are greater than CD, add DB to each of these; therefore the sides CE, EB are greater than CD, DB; but it has been shown that BA, AC are greater than BE, EC; much more then are BA, AC greater than BD, DC.

B

Again, because the exterior angle of a triangle is greater b 16. 1. than the interior and opposite angle, the exterior angle BDC of the triangle CDE is greater than CED; for the same reason, the exterior angle ČEB of the triangle ABE is greater than BAC; and it has been demonstrated that the angle