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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
a. Def. 1o.
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,
PROP. XIV. THEOR.
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
D Straight line with CE, let BE be
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.
PRO P. XV. THE OR.
opposite, angles shall be equal.
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.
PROP. XVI. THEOR.
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
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
2. 16. 1.
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.
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
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.
ANY two sides of a triangle are together greater
a. 3. I.
than the third side.
Produce PA to the point D,
Becaufe DA is equal to AC,
to ACD. but the angle BO!
c. 19. 1.
PROP. XXI. THEO R.
drawn two straight lines to a point within the
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