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Book 1.

Let the straight line AB make with CD, upon one lide 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

É
A

А.

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a def. 10. right a angle; but, if not, from the point B draw BE at right

angles to CD; therefore the angles CBE, EBD are two righe angles a; and because CBE is equal to the two angles CBA, ABE

together, add the angle EBD to each of these equals ; there. Ć 2. Ax. fore the angles CBE, EBD are equal c 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 de

monstrated to be equal to the fame three angles; and things d 1. Ax. that are equal to the same are equal d 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. 0. E. D.

PRO P. XIV. THLOR.
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 oppolite sides
of AB, make the adjacent angles
ABC, ABD equal together to
two right angles. BD is in the
same Itraight line with CB.

E
For, if BD be not in the same
straight line with CB, let BE be

B

D

,

in the fame straight line with it; therefore, because the straight Book I. line AB makes angles with the straight line CBE, upon one side of it, the angles ABC, ABE are together equal a to two a 13. I* sight angles ; 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 b to the remaining b 3. Ax. angle ABD, the less 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 Itraight line with CB. Wherefore, if at a point, &c. Q. E. D.

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IF.two straight lines cut one another, the vertical, or op

,

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 AŁD.

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

a 13. II Again, because the straight line DE makes with AB the angles A E

B AED, DEB, there also together equal a to two right angles; and CEA, AED have

D 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 to the remain.b 3. Ax. ing 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 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 consequently that all the angles made by any number of lines meeting in one point, are together equal to four right angles.

PROP.

IF.

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Book 1.

PRO P. XVI. THE OR one side 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
oppofte angles CBA, BAC.
Biļect a 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
EC, and BE to EF; AE, EB

E
are equal to CE, EF, each to

cach; and the angle AEB is bo IS. I. equalbto the angle CEF, be

B
C

D cause they are opposite ver

tical angles; therefore the C. 4 I. 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 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 ACD is greater than BAE: In the

fame manner, if the side BC be bisected, it may be demonstrated d. 15. 1. that the angle BCG, that is d, the angle ACD), is greater than

the angle ABC. Therefore, if one fide, &c. Q. E. D.

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A

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

А
Let ABC be any triangle ;
any 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 18
greater a than the interior and

B

C D opposite angle ABC; to each of

these

2. 16. I.

these add the angle ACB; therefore the angles ACD, ACB are Book I. greater than the angles ABC, ACB, but ACD, ACB are to gether equal b to two right angles; therefore the angles ABC 6.13. 1. BCA are less than two right angles. In like manner, it may demonstrated, that BAC, ACB, as also CAB, ABC are less than two right angles. Therefore any two angles, &c. Q. E. D.

be

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

.

Let ABC be a triangle, of which the fide 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 a AD equal to AB,

a. 3. I. and join BD; and because ADB is the exterior angle of the tri

B

C angle BDC, it is greater b than

b. 16. f. the interior and opposite angle DCB ; but ADB is equal to c. 5. I. ABD, because the Gde AB is 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.

PRO P. XIX. THE O R.
HE greater angle of every triangle is subtended by

the greater side, or has the greater fude 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 muft either be equal to AB, or

A less than it; it is not equal, because 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 less; because then the angle B

a. s. I.

ABC

Book 1. ABC would be less than the angle ACB; but it is not; there:

fore the side AC is not less than AB; and it has been thewn that it is not equal to AB; therefore AC is greater than AB. Wherefore the greater angle, &c. Q: E. D.

18. I.

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See N.

Abe third fide.

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

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

Produce B A to the point D, and make a AD equal to AC;

D and join DC.

Because DA is equal to AC, the angle ADC is likewise equal þ to ACD; but the angle BCD is greater than the angle ACD; therefore the angle BCD is great. B er than the angle ADC; and because the angle BCD of the triangle DCB is greater than its angle BDC, and that the greater side is opposite to the greater angle; therefore the side DB is greater than the side BC; but DB is equal to BA and AC; therefore the ides BA, AC are greater than BC. In the same manner it may be demonstrated, that the ldes AB, BC are greater than CA, and BC, CA greater than AB. Therefore any two sides, &c. Q. E. D.

C

C19 1.

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See N.

Il; 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 tri. angle, but shall contain a greater angle.

Let the two straight 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 sides of a triangle are greater than the third fide, the two sides BA, AE of the tri

angle

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