BC is equal to EF; therefore BC coinciding with EF; BA BOOK I. and AC shall coincide with ED and DF; for, if the base BC coincides with the base EF, but the sides BA, CA do not coincide with the sides ED, FD, but have a different situation as EG, FG, then, upon the same base EF, and upon the same side of it, there can be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise their sides terminated in the other extremity: But this is impossible; therefore, if the base BC * 7. 1. coincides with the base EF, the sides BA, AC cannot but coincide with the sides ED, DF; wherefore likewise the angle BAC coincides with the angle EDF, and is equal to `8 Ax. it. Therefore if two triangles, &c. Q. E. D.

PROP. IX. PROB.

To bisect a given rectilineal angle, that is to divide it into two equal angles.

t

Let BAC be the given rectilineal angle, it is required to bisect it.it Din A

Take any point Din AB, and from AC cut off AE equal to AD; join DE, and upon it describe an equilateral triangle DEF; then join AF; the straight line AF bisects the angle BAC.

Because AD is equal to AE, and AF is. common to the two triangles DAF, EAF; the two sides DA, AF, are equal to the two sides EA, AF, each to each; and the base DF is equal to the base EF; therefore the angle DAF is equal to the angle EAF: wherefore the given rectilineal

1

DA

A

angle BAC is bisected by the straight line AF. Which was to be done.

PROP. X. PROB.

To bisect a given finite straight line, that is, to divide it into two equal parts.

a

Let AB be the given straight line; it is required to divide it into two equal parts.

b

a 3. 1.

b 1.1.

c 8. 1.

Describe upon it an equilateral triangle ABC, and bisect 1. 1. the angle ACB by the straight line CD. AB is cut into two 9. 1. equal parts in the point D.

BOOK I.

Because AC is equal to CB, and CD common to the two triangles ACD, BCD; the two sides AC, CD are equal to BC, CD, each to each; and the angle ACD is equal to the angle BCD; there4. 1. fore the base AD is equal to the base DB, and the straight line AB is divided into two equal parts in the point D. Which was to be done.

NA

See N.

PROP. XI. PROB.

이

To draw a straight line at right angles to a given straight line, from a given point in the same.

Let AB be a given straight line, and Ca point given in it; it is required to draw a straight line from the point C at right angles to AB.

3.1. Take any point D in AC, and make CE equal to CD, and 1,1. upon DE describe the equi

b

lateral triangle DFE, and join
FC, the straight line FC drawn
from the given point C is at right
angles to the given straight line
AB.

Because DC is equal to CE,
and FC common to the two
triangles DCF, ECF; the two

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sides DC, CF, are equal to the two EC, CF, each to each; and the base DF is equal to the base EF; therefore the angle DCF 8. 1. is equal to the angle ECF; and they are adjacent angles. But, when the adjacent angles which one straight line makes with another straight line are equal to one another, each of 10 Def. them is called a right angle; therefore each of the angles DCF, ECF, is a right angle. Wherefore, from the given point C, in the given straight line AB, FC has been drawn at right angles to AB. Which was to be done.

COR. By help of this problem, it may be demonstrated, that two straight lines cannot have a common segment.

If it be possible, let the two straight lines ABC, ABD have the segment AB common to both of them. From the point B draw BE at right angles to AB; and because ABC is a straight

line, the angle CBE is equal a to
the angle EBA; in the same
manner, because ABD is a
straight line, the angle DBE is
equal to the angle EBA; where-
fore the angle DBE is equal to
the angle CBE, the less to the
greater; which is impossible; A
therefore two straight lines can-
not have a common segment.

PROP. XII. PROB

To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point 411 without it. sows sis of lampo at 100 su 5 ad bras

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C

Let AB be the given straight line, which may be produced to any length both ways, and let C be a point without it. It is required to draw a straight line perpendicular to AB from the point C.

Take any point D upon the other side of AB, and from the centre C, at the distance CD, describe the circle EGF meet-A ing AB in FG; and bisect FG in H, and join CF, CH, CG;

H

B

D

the straight line CH, drawn from the given point C, is perpendicular to the given straight line AB.

Because FH is equal to HG, and HC common to the two triangles FHC, GHC, the two sides FH, HC are equal to the two GH, HC, each to each; and the base CF is equal to the base CG; therefore the angle CHF is equal to the angle CHG; and they are adjacent are adjacent angles; but when a straight line standing on a straight line makes the di angles equal to

e

one another, each of them is a right angle; and the straight line which stands upon the other is called a perpendicular to it; therefore from the given point C a perpendicular CH has been drawn to the given straight line AB. Which was to be done.

PROP. XIII. THEOR.

THE angles which one straight line makes with another upon the one side of it, are either two right angles, or are together equal to two right angles.

BOOK I.

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

a

Def. 10. right angle; but, if not, from the point B draw BE at right 11. 1. angles to CD; therefore the angles CBE, EBD are two right

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angles; and because CBE is equal to the two angles CBA; ABE together, add the angle EBD to each of these equals; 2 Ax. therefore the angles CBE, EBD are equal to the three angles CBA, ABE, EBD. Again, because the angle DBA is be 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; 1 Ax. and things 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, 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. THEOR.

IF, 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 BC, BD upon the opposite sides of AB, make the adjacent angles ABC, ABD equal together to two right angles, BD is in the same straight line with CB.

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

A

E

D

in the same 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 to two a 13. 1. right 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 to the re- b 3. Ax. maining angle ABD, the less to the greater, which is impos to one sible; 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.

PROP. XV. THEOR.

IF 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 shall be equal to the angle DEB, and CEB to AED.

Because the straight line AE makes with CD the angles CEA, C AED, these angles are together equal to two right angles.Again, because the straight line A DE makes with AB the angles AED, DEB, these also are together equal to two right angles; and CEA, AED, have 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 to the remaining angle DEB. In the same manner 3. Ax. 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.

C