Again, because the exterior angle of a triangle is greater than the interior and opposite angle, the exterior angle BDC of the triangle CDE is greater than CED; for the same reason, the exterior angle CEB of the triangle ABE is greater than BAC; and it has been shewn that the angle BDC is greater than CEB; much more then is BDC greater than BAC.

XXII.—To make a triangle of which the sides shall be

equal to three given straight lines, but any two whatever of these must be greater than the third. Let A, B, C, be the three given straight lines, of which any two whatever are greater than the third viz., A and B greater than C; A and C greater than B; and B and C than A. It is required to make a triangle of which the sides shall be equal to A, B, C, each to each.

K

G

HE

B.

Take a straight line DE terminated at the point D, but unlimited towards E, and make DF equal to A, FG to B, and GH equal to C; and from the centre F, at the distance FD, describe the circle DKL; and from the centre G, at the distance GH, describe another circle HLK; and join KF, KG; the triangle KFG has its sides equal to the three straight lines, A, B, C.

Because the point F is the centre of the circle DKL, FD is equal to FK; but FD is equal to the straight line A; therefore FK is equal to A. Again, because G is the centre of the circle LKH, GĦ is equal to GK; but GH is equal to C; therefore also GK is equal to C; and FG is equal to B; therefore the three straight lines KF, FG, GK, are equal to the three A, B, C: and therefore the triangle KFG has its three sides KF, FG, GK, equal to the three given straight lines A, B, C.

XXIII.-At a given point in a given straight line, to make a rectilineal angle equal to a given rectilineal angle.

Let AB be the given straight line, and A the given point in it, and DCE the given rectilineal angle; it is

required to make an angle at the given point A in the given straight line AB, that shall be equal to the given rectilineal angle DCE.

B

F

Take in CD, CE, any points D, D E, and join DE; and make the triangle AFG, the sides of which shall be equal to the three straight lines CD, DE, EC, so that CD be equal to AF, CE to AG, and DE to FG; and because DC, CE are equal to FA, AG, each to each, and the base DE to the base FG; the angle DCE is equal to the angle FAG. Therefore, at the given point A in the given straight line AB, the angle FAG is made equal to the given rectilineal angle DCE.

XXIV.-If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other; the base of that which has the greater angle shall be greater than the base of the other.

Let ABC, DEF be two triangles which have the two sides, AB, AC, equal to the two DE, DF, each to each, viz., AB equal to DE, and AC to DF; but the angle BAC greater than the angle EDF; the base BC is also greater than the base EF.

Of the two sides DE, DF, let DE be the side which is not greater than the other, and at the point D, in the straight line DE, make the angle EDG equal to BAC; and make DG equal to AC or DF, and join EG, GF.

Because AB is equal to DE, and AC to DG, the two sides BA, AC, are equal to the two ED, DG, each to each, and the angle A

BAC is equal to EDG; there

D

fore BC is equal to EG; and

because DG is equal to DF,

the angle DFG is equal to B

DGF; but DGF is greater

than EGF; therefore DFG is greater than EGF; and

much more is EFG greater than EGF; and because the angle EFG of the triangle EFG is greater than its angle EGF, and that the greater side is opposite to the greater angle; the side EG is therefore greater than the side EF; but EG is equal to BC; and therefore also BC is greater than EF.

XXV.—If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of the one greater than the base of the other; the angle also contained by the sides of that which has the greater base, shall be greater than the angle contained by the sides equal to them of the other. Let ABC, DEF, be two triangles which have the two sides AB, AC, equal to the two sides DE, DF, each to each, viz., AB equal to DE, and AC to DF; but the base CB greater than the base EF; the angle BAC is likewise greater than the angle EDF.

B

D

E

For if it be not greater it must either be equal to it, or less; but the angle BAC is not equal to EDF, because then the base BC would be equal to EF; but it is not; therefore the angle BAC is not equal to EDF; neither is it less; because then the base BC would be less than EF; but it is not; therefore the angle BAC is not less than EDF; and it was shown that it was not equal to it; therefore the angle BAC is greater than EDF.

XXVI.-If two triangles have two angles of one equal

to two angles of the other, each to each; and one side equal to one side, viz., either the sides adjacent to the equal angles, or the sides opposite to equal angles in each; then shall the other sides be equal each to each; and also the third angle of the one to the third angle of the other.

Let ABC, DEF, be two triangles which have the angles ABC, BCA, equal to DEF, EFD; viz., ABC to DEF, and BCA to EFD; also one side equal to one

E

side; and first let those sides be equal which are adjacent to the angles that are equal in the two triangles, viz., BC to EF; the other sides shall be equal, each to each, viz., AB to DE, and AC to DF; and the third angle BAC to the third angle EDF. For, if AB be not equal to DE, one of them must be the greater. Let AB be the greater, and make BG equal to DE, and join GC; therefore, because BG is equal to DE, and BC to EF, the two sides GB, BC, are equal to the two, DE, EF, each to each; and the angle GBC is equal to DEF; therefore the base GC is equal to DF, and the triangle GBC to DEF, and the other angles to the other angles, each to each, to which the equal sides are opposite; therefore the angle GCB is equal to DFE; but DFE is, by the hypothesis, equal to BCA; wherefore also the angle BCG is equal to BCA, the less to the greater, which is impossible; therefore AB is not unequal to DE, that is, it is equal to it; and BC is equal to EF; therefore the two, AB, BC, are equal to the two DE, EF, each to each; and the angle ABC is equal to DEF; therefore the base AC is equal to DF, and the third angle BAC to the third angle EDF.

A

B

H C E

Next, let the sides which are opposite to equal angles in each triangle be equal to one another, viz., AB to DE; likewise in this case, the other sides shall be equal, AC to DF, and BC to EF; and also the third angle BAC to the third EDF.

the

For, if BC be not equal to EF, let BC be the greater of them, and make BH equal to EF, and join AH; and because BH is equal to EF, and AB to DE, two AB, BH, are equal to the two DE, EF, each to each; and they contain equal angles; therefore the base AH is equal to DF, and the triangle ABH tc DEF, and the other angles shall be equal, each to each, to which the equal sides are opposite; therefore the gle BHA is equal to the angle EFD; but EFD is

equal to the angle BCA; therefore also the angle BHA is equal to the angle BCA, that is, the exterior angle BHA of the triangle AHC is equal to its interior and opposite angle BCA, which is impossible; wherefore BC is not unequal to EF, that is, it is equal to it; and AB is equal to DE; therefore the two AB, BC, are equal to the two DE, EF, each to each; and they contain equal angles; wherefore the base AC is equal to the base DF, and the third angle BAC to the third angle EDF.

XXVII.-If a straight line falling upon two other

straight lines makes the alternate angles equal to one another, these two straight lines shall be parallel. Let the straight line EF, which falls upon the two straight lines AB, CD, make the alternate angles AEF, EFD, equal to one another; AB is parallel to CD.

For, if it be not parallel, ÁB and CD being produced shall meet either towards B, D, or towards A, C; let them be produced and meet towards

B, D, in the point G; therefore AE B
GEF is a triangle, and its exterior
angle AEF is greater than the in- C F
terior and opposite angle EFG; but

it is also equal to it, which is impossible; therefore AB and CD being produced do not meet towards B, D. In like manner it may be demonstrated that they do not meet towards A, Č; but those straight lines which meet neither way, though produced ever so far, are parallel to one another. AB therefore is parallel to CD.

XXVIII.—If a straight line, falling upon two other straight lines makes the exterior angle equal to the interior and opposite upon the same side of the line; or makes the interior angles upon the same side together equal to two right angles; the two straight lines shall be parallel to one another.

Let the straight line EF, which falls upon the two straight lines AB, CD, make the exterior angle EGB equal to the interior and opposite angle GHD) upon the same side; or make the interior angles on the