Book I.

Let the ftraight line AB make with CD, upon one fide of it, the angles CBA, ABD; thefe 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

b II. I.

c 2. Ax.

a def. 10. right angle; but, if not, from the point B draw BE at right angles to CD; therefore the angles CBE, EBD are two right angles a; and becaufe 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 equal to the three angles CBA, ABE, EBD. Again, because the angle DBA is equal to the two angles DBE, EBA, add to thefe 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 d to one another; therefore the angles CBE, LBD 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 ftraight line, &c. Q. E. D.

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

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

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

A.

E

B

D

in the fame ftraight line with it; therefore, because the straight Book I. line AB makes angles with the ftraight line CBE, upon one fide of it, the angles ABC, ABE are together equal a to two a 13. I right angles; 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 b 3. Ax. angle ABD, the lefs to the greater, which is impoffible; there fore BE is not in the fame ftraight 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.

PROP. XV. THE OR.

IF two ftraight lines cut one another, the vertical, or op‐ pofite, angles fhall be equal.

Let the two ftraight lines AB, CD cut one another in the point E; the angle AEC fhall be equal to the angle DEB, and CEB to ALD.

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 A AED, DEB, these alfo are together equal a to two right angles; and CEA, AED have been demonftrated to be equal to

E

B

D

two right angles; wherefore the angles CEA,

b

AED are equal

a 13. 1

to the angles AED, DEB. Take away the common angle AED, and the remaining angle CEA is equal to the remain-b 3. Ax. ing angle DEB. In the fame manner it can be demonstrated that the angles CEB, AED are equal. Therefore, if two ftraight lines, &c. Q. E. D.

COR. 1. From this it is manifeft, that, if two ftraight 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.

PROP.

Book I.

PRO P. XVI.

THEOR.

a. 10. I.

IF one fide 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 fide BC be produced to D, the exterior angle ACD is greater than either of the interior oppofite angles CBA, BAC.

Bifect a AC in E, join BE
and produce it to F, and make
EF equal to BE; join alfo
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. 15. I equal to the angle CEF, because they are oppofite vertical angles; therefore the bafe AB is equal to the bafe CF, and the triangle AEB to the triangle CEF, and the

C. 4. I.

d. 15. f.

2. 16. 1.

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 demonstrated that the angle BCG, that is d, the angle ACD, is greater than the angle ABC. Therefore, if one fide, &c. Q. E. D.

A

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

thefe add the angle ACB; therefore the angles ACD, ACB are Book I. greater than the angles ABC, ACB; but ACD, ACB are to- in gether equal b to two right angles; therefore the angles ABC b. 13. 1. HBCA are less than two right angles. In like manner, it may be demonftrated, that BAC, ACB, as also CAB, ABC are less than two right angles. Therefore any two angles, &c. Q. E. D.

PROP. XVIII. THEOR.

HE greater fide of every triangle is oppofite to the

T greater angle.
THE

Let ABC be a triangle, of which the fide AC is greater than the fide AB; the angle ABC is also 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 triangle BDC, it is greater b than

the interior and oppofite angle DCB; but ADB is equal e to c. §. I. ABD, because the fide AB is equal to the fide AD; therefore the angle ABD is likewife greater than the angle ACB; wherefore much more is the angle ABC greater than ACB. Therefore the greater fide, &c. Q. E. D.

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

THE

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
muft either be equal to AB, or
lefs 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
lefs; because then the angle B

A

a. 5. I.

ABC

ABC would be lefs b than the angle ACB; but it is not; there: fore the fide AC is not less than AB; and it has been the wn that it is not equal to AB; therefore AC is greater than AB. Wherefore the greater angle, &c. Q· E. D.

A

NY 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; therefore the angle BCD is greater than the angle ADC; and be

с

B

D

cause 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, from the ends of the fide of a triangle, there be drawn two ftraight lines to a point within the triangle, thefe fhall be lefs 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 less 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 because two fides of a triangle are greater than the third fide, the two fides BA, AE of the triangle