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Book 1.

Let the straight 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 A

E

A

a. Def. 1o.

DB

B b. 11. 1. angle. but if not, from the point B draw BE at right anglesbto 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 these equals, therefore the angles CBE, EBD are C. 2. Ax. 'equal < to the three angles CBA, ABE, EDD. 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 demonstrated to be equal to the same three angles; and things d. 1. Ax. 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,
EBDare 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.
IF
F at a point in a straight line, two other straight

lincs, upon the opposite sides of it, make the adjacent angles together equal to two right angles, thesetwo straight lines thall 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 fides
of AB, make the adjacent angles
ABC, ABD equal together to
two right angles. BD is in the

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

D Straight line with CE, let BE be

B

Book lo in the fame Araight line with it. therefore because 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; a 13. . but the 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 b. 3. An. greater, which is imposible. 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.

IF

PRO P. XV. THE OR.
F 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 Thail be equal to the angle DEB, and CEB to AED.

Because the straight line AE makes with CD the angles CEA, AED, these angles are together equal to two right angles. again,

2. 13. to because the straight line DE makes with AB the angles AED, DEB ; these also are together equal to two right angles. and CEA, AED A E

8 have been demonstrated to be equal 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 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 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.

B

Book 1.

VIF

PROP. XVI. THEOR.
F one side of a triangle be produced, the exterior

argle 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 opposite angles CBA, BAC. BifectAC in E, join BE

A
and produce it to F, and make

F
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 A EB is B

D
equal b to the angle CEF, be-
cause they are opposite verti-
cal angles. therefore the base
AB is equal to the base
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 opposite. wherefore the angle BAE is equal to the angle
ECF. but the angle ECD is greater than the angle ECF, there-
fore the angle ACD is greater than BAE. in the same manner,
if the side L'C be bifected, it may be demonstrated that the angle
BCG, that is 4, th angle ACD, is greater than the angle ABC.
therefore if one sile, &c. Q. E. D.

PROP. XVII. THEO R.

b. 15. s.

C. 4. I.

d. 15. 1.

ANY two angl s of a triangle are together less 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 be. cause ACD is the exterior antle of the triangle ABC, ACD is greater · than the interior an! opposite angle ABC; to each of

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these add the angle ACB, therefore the angles ACD, ACB are Book 1. greater than the angles ABC, ACB. but ACD, ACB are together equal o to two right angles ; therefore the angles ABC, ECA are b. zz. to 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.

TH

THE greater side of every triangle is opposite to the

greater angle. Let ABC be a triangle of

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

PROP. XIX. THE O R.

THE greater angle of every triangle is fubtended by

THE

of 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 fide 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 co qual' to the angle ACB ; but it

2. $. . is not; therefore AC is not equal to AB. neither is it less; because.

B then the angle ABC would be less

Book 1. 6 than the angle ACB ; but it is not; therefore the fide AC is

m not less than AB. and it has been shewn that it is not equal to b. 18. 1. AB. therefore AC is greater than AB. wherefore the greater

angle, &c. Q. E. D.

PRO P. XX.

THE O R.

Sec N.

ANY two sides of a triangle are together greater

a. 3. I.

b. s.1.

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

Produce PA to the point D,
and make • AD equal to AC,
and join DC.

Becaufe DA is equal to AC,
the angle ADC is likewife equal

to ACD. but the angle BO!
is greater than the angle ACD; B
therefore the angle PCD is great-
er than the angle ADC. and be afe the angle BCD of the tri-
angle DCB is greater than its angle BDC, and that the greater
fide is opposite to the gr ater angle, therefore the side DB is greater
than the side 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 files, &c. Q. E. D.

b

c. 19. 1.

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PROP. XXI. THEO R.
IF F 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 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 triangle are greater than the third fide, the two fides BA, AE of the triangle ABE

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