BOOK I. CASE II. Next, let the sides which are opposite to equal angles,

b xvi. 1.

and also the third angle BAC to the third EDF.

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;

the 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 the base DF,

and the triangle ABH to the triangle DEF,

and the other angles shall be equal,

each to each, to which the equal sides are opposite;
therefore the angle 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 b;

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.

Therefore, if two triangles, &c. Q. E. D.

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,

AB 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 GEF is a triangle,

a

and its exterior angle AEF is greater than the interior and

but those straight lines which meet neither way,

though produced ever so far,

are parallel to one another.

BOOK I.

a xvi. 1

b 35 Def.

AB therefore is parallel to CD.

Wherefore, if a straight line, &c. Q. E. D.

E

BOOK L

PROP. XXVIII. THEOR.

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

a XV. 1.

(1.) Because the angle EGB is equal to the angle GHD,

a

and the angle EGB equal to the angle AGH,

the angle AGH is equal to the angle GHD;

and they are the alternate angles;

xxvii. 1. therefore AB is parallel to CD.

(2.) Again, because the angles BGH, GHD

By Hyp. are equal to two right angles;

d xiii. 1. and that AGH, BGH are also equal to two right angles;

the angles AGH, BGH are equal to the angles BGH, GHD.

Take away the common angle BGH;

therefore the remaining angle AGH is equal to

the remaining angle GHD;

and they are alternate angles ;

therefore AB is parallel to CD.

Wherefore if a straight line, &c. Q. E. D.

PROP. XXIX. THEOR.

If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another.

And the exterior angle equal to the interior and opposite upon the same side.

And likewise the two interior angles upon the same side together equal to two right angles.

Let the straight line EF fall upon

the parallel straight lines AB, CD;

(1.) the alternate angles AGH, GHD are equal to one another;

(2.) and the exterior angle EGB is equal to the interior and opposite, upon the same side GHD;

(3.) and the two interior angles

E

BOOK I.

(1.) For, if AGH be not equal to GHD;

one of them must be greater than the other;

let AGH be the greater;

and because the angle AGH is greater than the angle GHD; add to each of them the angle BGH;

therefore the angles AGH, BGH are greater than the

angles BGH, GHD;

but the angles AGH, BGH are equal to two right angles;
therefore the angles BGH, GHD are less than two right angles;
but those straight lines which, with another straight line

falling upon them, make the interior angles on the same
side less than two right angles, do meet together if
continually produced;

therefore the straight lines AB, CD, if produced far enough,
shall meet;

but they never meet;

since they are parallel by the hypothesis ;

therefore the angle AGH is not unequal to the angle GHD;

a xiii. 1

BOOK I. that is, it is equal to it;

xv. 1.

• xiii. 1.

(2.) but the angle AGH is equal to the angle EGB;
therefore likewise EGB is equal to GHD:

(3.) add to each of these the angle BGH;
therefore the angles EGB, BGH are equal
to the angles BGH, GHD;

с

but EGB, BGH are equal to two right angles;

therefore also BGH, GHD are equal to two right angles. Wherefore, if a straight, &c. Q. E. D.

Straight lines which are parallel to the same straight line are

parallel to each other.

Let AB, CD be each of them parallel to EF;

AB is also parallel to CD.

Let the straight line GHK cut AB, EF, CD;

cuts the parallel straight lines EF, CD,

the angle GHF is equal to the angle GKD;

and it was shown that the angle AGK

is equal to the angle GHF;

therefore, also, AGK is equal to GKD;
and they are alternate angles;

xxvii. 1. therefore AB is parallel to CD.

Wherefore straight lines, &c. Q. E. D.