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Book I. Let the straight line AB make with CD, upon one side of it, the Mangles CBA, ABD; these are either two right angles, or are toge

ther equal to two right angles. a. Def. 10. For if the angle CBA be equal to ABD, each of them is a right

E
A

Α.

d. 1. Ax.

D
B

D

B b. 11.1. angle. but if not, from the point B draw BE at right angles b to CD.

therefore the angles CBE, EBD are two right anglesa, and because CBE is equal to the two angles CBA, ABE together ; add the angle

EBD to each of these equals, therefore the angles CBE, EBD are c. 2. Ax. equal to the three angles CBA, ABE, EBD, again, because the

angle DBA is equal to the two angles DBE, EBA, add to these e-
quals the angle ABC; therefore the angles DBA, ABC are equal to
the three angles DBE, EBA, ABC. but the angles CBE, EBD have
been demonstrated to be equal to the same three angles; and things
that are equal to the same are equal to one another; 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.
F 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,

A.
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

E
straight line with CB.
For if BD be not in the same

C B

D straight line with CB, let BE be

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in the same straight line with it. therefore because the straight line Book I. AB makes angles with the straight line CBE, uport one side of it, m the angles ABC, ABE are together equaloto two right angles; but the 2. 13. 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 b. 3. Ax. 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 fame straight line with CB, Wherefore if at a point, &c. Q: E. D.

IF

PROP. XV. THEOR. 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 åre together equal to two right angles, again,

2. 13. 1. because the straight line DE makes with AB the angles AED, DEB; these also are together e

E qual a to two right angles. and A

B 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 o b. 3. Ax. 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 manifelt 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.

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.

I

1. 10. 1.

G

C. 4. I.

Book 1.

PROP. XVI. THEOR
F one Gide 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 op-
posite angles CBA, BAC.
Bisect * AC in E, join BE

A
and produce it to F, and make
EF equal to BE; join also
FC, and produce AC to G.
Because AE is equal to

E
EC, and BE to EF; AE, EB
are equal to CE, EF, each
to each; and the angle AEB B

C

D 6.15. 1. is equal b to the angle CEF,

because they are opposite ver-
tical angles. therefore the
bafe AB is equal to the
base CF, and the triangle AEB to the triangle CEF, and the re-
maining angles to the remaining angles, each to each, to which the
equal fides 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 man-

ner, if the side BC be bisected, it may be demonstrated that the d. 15.1.' angle BCG, that is d, the angle ACD, is greater than the angle ABC. therefore if one side, &c. Q. E. D.

PROP. XVII. THEOR.
NY two angles of a triangle are together less than

two right angles.
Let ABC be any triangle; any

A
two of its angles together are less
than two right angles,

Produce BC to D; and be.
cause ACD is the exterior angle

of the triangle ABC, ACD is &. 1$. 1. greater

a than the interior and opposite angle ABC; to each of

B

С D

these

A

19

these add the angle ACB, therefore the angles ACD, ACB are great- Book I. er than the angles ABC, ACB. but ACD, ACB are together equalb m to two right angles; therefore the angles ABC, BCA are less than b. 13. 1. 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

HE greater side of every triangle is opposite to the

greater angle. Let ABC be a triangle of which the side AC is greater

А than the side AB; the angle ABC is also greater than the angle BCA. Because AC is greater than

D AB, make · AD equal to AB,

a. h.ts and join BD, and because ADB

B is the exterior angle of the triangle BDC, it is greater than the interior and opposite angle b. 16. 1: DCB. but ADB is equal o to ABD, because the side AB is e- c. s. i. qual to the side AD; therefore the angle ABD is likewise greater than the angle ACB; wherefore much more is the angle ABC greattr than ACB. therefore the greater fide, &c. Q. E. D.

T

PROP. XIX. THEOR.
HE

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

A 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

8. $.fi is not; therefore AC is not equal to AB. neither is it less; because B then the angle ABC would be less B2

than

'Book 1. than the angle ACB; but it is not; therefore the side AC is not

less than AB, and it has been shewn that it is not equal to AB. therefore AC is greater than AB. wherefore the greater angle, &c, Q. E. D.

b 18. 1.

PROP. XX. THEOR.

Sec N.

A

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NY 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,

D
and make · AD equal to AC, and
join DC.

Because DA is equal to AC,
the angle ADC is likewise equal
b to ACD. but the angle BCD
is greater than the angle ACD; B

с
therefore the angle BCD is great-
er than the angle ADC, and because the angle BCD of the tri-
angle DCB is greater than its angle BDC, and that the greater
fide is opposite to the greater angle, therefore the side DB is greater
than the fide BC, but DB is equal to BA and AC; therefore the
sides BA, AC 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.

b.g.1.

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C. 19. 1.

PROP. XXI. THEOR.

Sec N.

TF from the ends of the side of a triangle there be drawn

two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but Thall 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 triangle are greater than the third fide, the two sides BA, AE of the triangle ABE

are

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