Book I.

a. Def. 10.

Let the ftraight line AB make with CD, upon one side of it, the angles CBA, ABD; these are either two right angles, or are together equal to two right angles.

For if the angle CBA be equal to ABD, each of them is a right

b... angle. but if not, from the point B draw BE at right angles to CD. therefore the angles CBE, EBD are two right angles". and because CBE is equal to the two angles CBA, ABE together; add the angle EBD to each of thefe equals, therefore the angles CBE, EBD are

c

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 equals the angle ABC; therefore the angles DBA,ABC are equal to the three angles DBE, EBA, ABC. but the angles CBE, EBD have been demonftrated to be equal to the fame three angles; and things d. 1. Ax. that are equal to the fame 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.

I'

THEOR.

Fat a point in a straight line, two other straight lines, upon the oppofite fides of it, make the adjacent angles together equal to two right angles, these two ftraight lines fhall be in one and the fame straight line.

At the point B in the straight line AB, let the two ftraight lines, BC, BD upon the oppofite fides of AB, make the adjacent angles ABC, ABD equal together to two right angles. BD is in the fame ftraight line with CB.

For if BD be not in the fame ftraight line with CE, let BE be

A

E

C

B

D

Book 1.

in the fame Araight line with it. therefore because the straight line AB makes angles with the straight line CBE, upon one fide of it, the angles ABC, ABE are together equal to two right angles; a 13. 1. but the angles ABC, ABD are likewife 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 lefs to the b. 3. A greater, which is impoffible. therefore BE is not in the fame straight line with BC. And in like manner, it may be demonftrated that no other can be in the fame ftraight line with it but BD, which therefore is in the fame ftraight line with CB. Wherefore if at a point, &c. Q. E. D.

b

IF

F two ftraight lines cut one another, the vertical, or oppofite, angles fhall be equal.

Let the two ftraight lines AB, CD cut one another in the point E. the angle AEC fhail be equal to the angle DEB, and CEB to AED. Because the ftraight line AE makes with CD the angles CEA,

AED, thefe angles are together equal to two right angles. again, because the straight line DE makes with AB the angles AED, DEB; thefe alfo are together equal to

two right angles. and CEA, AED A

E

B

have been demonftrated to be e

qual to two right angles; where

D

fore 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 b. 3. As. angle DEB. In the fame manner it can be demonftrated that

the angles CEB, AED are equal. therefore if two straight lines, &c. Q. E. D.

COR. 1. From this it is manifeft 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 confequently that all the angles made by any number of lines meeting in one point, are together equal to four right angles.

B

a. 16. I.

IF

PROP. XVI. THEOR.

Fone fide of a triangle be produced, the exterior angle is greater than either of the interior oppofite angles.

Let ABC be a triangle, and let its fide BC be produced to D. the exterior angle ACD is greater than either of the interior op posite angles CBA, BAC.

A

F

4

Bifect AC in E, join BE
and produce it to F, and make
EF equal to BE; join also
FC, and produce AC to G.
Because AE is equal to
EC, and BE to EF; AE, EB
are equal to CE, EF, each to
each; and the angle AEB is B
equal to the angle CEF, be-
caufe they are oppofite verti-
cal angles. therefore the bafe
AB is equal to the bafe

c

D

CF, and the triangle AEB to the triangle CEF, and the remaining angles, to the remaining angles, each to each, to which the equal fides are oppofite. 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 fame manner, if the fide BC be bifected, it may be demonftrated that the angle BCG, that is, the angle ACD, is greater than the angle ABC. therefore if one fide, &c. Q. E. D.

A

PROP. XVII. THEOR.

NY two angls of a triangle are together lefs than two right angles.

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

Produce BC to D; and becaufe ACD is the exterior angle of the triangle ABC, ACD is greater than the interior an' eppofite angle ABC; to each of

A

D

C

32

thefe add the angle ACB, therefore the angles ACD, ACB are Book I. greater than the angles ABC, ACB. but ACD, ACB are together

b

equal to two right angles; therefore the angles ABC, BCA are b. 13. 1.
lefs than two right angles. in like manner it may be demonftrated
that BAC, ACB, as also CAB, ABC are less than two right an-
gles. therefore any two angles, &c. Q. E. D.

THE

PROP. XVIII. THE OR.

greater fide of every triangle is opposite to the greater angle.

Let ABC be a triangle of which the fide AC is greater than the fide AB; the angle ABC is alfo greater than the angle BCA.

Because AC is greater than AB, make AD equal to AB, and join BD. and because ADB is the exterior angle of the tri

b

c

2. 3. 1

angle BDC, it is greater than the interior and oppofite angle. 16.
DCB. but ADB is equal to ABD, because the fide AB is c. 5. 1.
equal to the fide 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 fide, &c. Q. E. D.

PROP. XIX. THEOR.

THE greater angle of every triangle is fubtended by
the greater fide, or has the greater fide oppofite to it.

Let ABC be a triangle of which the angle ABC is greater than
the angle BCA. the fide AC is likewife greater than the fide AB.
For if it be not greater, AC

a

muft either be equal to AB, or lefs
than it. it is not equal, because
then the angle ABC would be e-
qual to the angle ACB; but it
is not; therefore AC is not equal
to AB. neither is it lefs; because.
B
then the angle ABC would be less

2. 5. f.

Book I. than the angle ACB; but it is not; therefore the fide AC is not lefs than AB. and it has been fhewn that it is not equal to AB. therefore AC is greater than AB. wherefore the greater angle, &c. Q. E. D.

b. 18. 1.

See N.

2. 3. I.

b. 5.1.

C. 19. I.

See N.

PROP. XX.

THEOR.

ANY two fides of a triangle are together greater

than the third fide.

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

Produce BA to the point D,

and make AD equal to AC,

and join DC.

Because DA is equal to AC,
the angle ADC is likewife equal
b to ACD. but the angle BCD
is greater than the angle ACD; B
therefore the angle PCD is great-

D

A.

C

er than the angle ADC. and becafe the angle BCD of the triangle DCB is greater than its angle BDC, and that the greater fide is oppofite to the greater angle, therefore the fide DB is greater than the fide BC. but DB is equal to BA and AC; therefore the fides BA, AC are greater than BC. in the fame manner it may be demonstrated that the fides AB, BC are greater than CA; and BC, CA greater than AB. therefore any two fides, &c. Q. E. D.

IF

PROP. XXI. THEOR.

F from the ends of the fide of a triangle there be drawn two ftraight lines to a point within the triangle, these fhall be less than the other two fides of the triangle, but fhall contain a greater angle.

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

Produce BD to E; and becaufe two fides of a triangle are greater than the third fide, the two fides BA, AE of the triangle ABE