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+ Hyp.

5.1.

+ Hyp.

7.1.

* 8 Ax.

*

BCD. Again, because CB is equal+ to DB, the angle BDC is equal to the angle BCD; but BDC has been proved to be greater than the same BCD; which is impossible. The case in which the vertex of one triangle is upon a side of the other, needs no demonstration.

Therefore, upon the same base, and on the same side of it, there cannot be two triangles that have their sides, which are terminated in one extremity of the base, equal to another, and likewise those which are terminated in the other extremity. Q. E. D.

PROP. VIII. THEOR.

If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal; the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides equal to them, of the other.

Let ABC, DEF be two triangles, having the two sides AB, AC, equal to the two sides DE, DF, each to each, viz. AB to DE, and AC to DF; and also the base BC equal to the base EF. The angle BAC shall be equal to the angle EDF.

For, if the triangle ABC be applied to DEF, so that the point B may be on E, and the straight line BC upon

A

B

DG

CE

EF; the point C shall also coincide with the point F, because BC is equal+ to EF. Therefore BC coinciding with EF, BA 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 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 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.

Let BAC be the given rectilineal angle; it is required to bisect it.

3. 1.

Take any point D in AB, and from AC cut* off AE equal to AD; join DE, and upon it describe an equila * 1. 1. teral triangle DEF; then join AF: the straight line AF shall bisect the angle BAC.

A

A

B F

+ Constr.

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 +Constr. the angle DAF is equal to the angle EAF: wherefore 8. 1. the given rectilineal 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.

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

1.1.

AB

9.1.

+ Constr.

Describe* upon it an equilateral triangle ABC, and bisect the angle ACB by the straight line CD. shall be cut into two equal parts in the point D. 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; therefore 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.

PROP. XI. PROB.

A D

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

+ Constr.

* 4.1.

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

* 3. 1.

* 1. 1.

+ Constr.

+ Constr. 8. 1.

Take any point D in AC, and make* CE equal to
CD, and upon DE describe* the equilateral triangle
DFE, and join FC. The straight line FC drawn from
the given point C, shall be at right

angles to the given straight line
AB.

*

A D

F

E B

Because DC is equal+ to CE, and FC common to the two triangles DCF, ECF; the two 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 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 them is * 10 Def. 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.

† 11. 1.

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

* 10 Def. angle CBE is equal* to the angle EBA;
in the same manner, because ABD is a
straight line, the angle DBE is equal to
the angle EBA; wherefore+ the angle
DBE is equal to the angle CBE, the A
less to the greater; which is impossible:

† 1 Ax.

B

D

therefore two straight lines cannot 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 without it.

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.

AF

C

H

*

* 10. 1.

Take any point D upon the other side of AB, and from the centre C, at the distance CD, describe the 3 Post. circle EGF, meeting AB in F, G; bisect* FG in H, and join CH. The straight line CH, drawn from the given point C, shall be perpendicular to the given straight line AB.

*

*

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15 Def. 8. 1.

Join CF, CG: and because FH is equal to HG, and † Constr. 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 angles: but when a straight line, standing on another straight line, makes the adjacent angles equal to one another, each of them is a right angle, and the straight line which stands upon the other, is called a perpendicular+ to it: there † 10 Def. fore, 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 one side of it, are either two right angles, or are together equal to two right angles.

Let the straight line AB make with CD, upon one side of it, the angles CBA, ABD: these shall either be two right angles, or shall together be equal to two right angles.

+ 10 Def.

E

A

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For if the angle CBA be equal to ABD, each of them is a 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: and because

D

B

D

B

*

* 10 Def.

the angle 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 equal to the three * 2 Ax. angles CBA, ABE, EBD. Again, because the angle DBA is equal to the two angles DBE, EBA, add to each of these equals the angle ABC; therefore the angles DBA, ABC are equal + to the three angles DBE, EBA, † 2 Ax. ABC: but the angles CBE, EBD have been demonstrated to be equal to the same three angles; and things

* 1 Ax.

† 1 Ax.

* 13. 1. † Hyp.

† 1 Ax.

* 3 Ax.

13. 1.

that are equal to the same thing, 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 shall be in the same straight line with CB.

*

C B

A

E

D

For, if BD be not in the same straight line with CB, let BE be in the same straight line with it: therefore, because the straight line AB makes with the straight line CBE, upon one side of it, the angles ABC, ABE, these angles are together equal to two 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, and the remaining angle ABE is equal to the remaining 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 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, AED, these angles are together equal to

*

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