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let BE be in the same straight line with it; therefore, be- Book I.
cause the straight line AB makes angles with the straight
line CBE, upon one side of it, the angles ABC, ABE are
together equala 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 equalb 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, 1 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.

For the angles CEA,
AED, which the straight
line AE makes with the
straight line CD, are to-
gether equal a to
to two A-

B a 13. 1.

E
right angles; and the an-
gles AED, DEB, which
the straight line DE makes
with the straight line AB,

'D
are also together equala 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 equalb to the remaining angle DEB. In b 3. Ax.
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.

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Book 1.

Monely

PROP. XVI. THEOR.

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IF one side of a triangle be produced, the exterior angle is greater than either of the interior and oppo. site 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, А.

F
BAC.

Bisecta ĄC in E, join BE
and produce it to F,

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

E
G.

Because AE is equal to EC,
and BE to EF, AE, EB are
equal to CE, EF, each to B

D each; and the angle AEB is equal to the - angle CEF, because they are vertical angles; therefore the base AB is equal c to the base CF,

G 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 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.

b 15. 1.

C 4. 1.

PROP. XVII. THEOR.

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

Let ABC be any triangle ;

Book I.

A
any two of its angles toge-
ther are less than two right
angles.

Produce BC to D; and
because ACD is the exterior
angle of the triangle ABC,
ACD is greater a than the

a 16. 1. interior and opposite angle ABC; to each of these add the angle ACB; therefore B

C

D 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.

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

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

D
From AC, which is greater than
AB, cut offa AD equal to AB, B
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 c 5.1.
AĎ 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. D.

Ċ a 3. 1.

caus

Book I.

PROP. XIX. THEOR.

ang gre BC

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

ani iti tha sid

to it.

be

a 5. 1.

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, be-
cause then the angle ABC would
be equal a to the angle ACB;
but it is not; therefore AC is
not equal to AB; neither is it

B
less; because then the 'angle
ABC would be lessb 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.

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b 18. 1.

PROP. XX. THEOR.

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

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a 3. 1.

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-
wisé equall to ACD; but
the angle BCD is greater
than the angle ACD; there- B

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

b 5. 1.

cause the angle BCD of the triangle DCB is greater than its Book I. angle BDC, and that the greater 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.

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

А. 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

C Again, because the exterior angle of a triangleb 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 CEB of the triangle ABE is greater than BAC; and it has been demonstrated that the angle

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