PROP. XXVI. THEOR. 27. 1 Eu.
If a straight line falling upon two other

straight lines, make the alternate angles
equal to one another, these two straight lines
shall be parallel.

Let str. line EF, which falls upon the str. lines AB, CD, make alt. LAEF = alt. ZEFD: then shall AB || CD.

For if AB H CD, they will, when produced, meet either towards B, D, or towards A, C. Def. 35.

Suppose the former, and let them meet in G, then GEF will form a A; .. ext. L AEF > int. _ EFG;

Prop. 15. but L AEF = _ EFG or _EFD, Hyp.

which is impossible. ... AB, CD, being produced, do not meet to

wards B, D. In like manner it may be proved that they

do not meet towards A, C;

.. AB || CD; Wherefore if a str. line, &c.

Def. 35.

PROP. XXVII. THEOR. 28. 1 Eu.

If a straight line falling upon two other

straight lines, make the exterior angle equal to the interior and opposite upon the same side of the line; or make the interior angles upon the same side together equal to two right angles ; the two straight lines shall be parallel to one another.

from these equals take BGH,
CİD-J LAGH, which are Ax. 3.

Prop. 26. : AB || CD. Wherefore, if a straight line, &c.

PROP. XXVIII. THEOR. 29. I Eu. 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 angle on the same side ; and likewise the two interior angles upon the same side together equal to two right angles.

Let the str. line EF fall on the str. lines AB, CD, then

alt. L AGH = alt. _ GHD,

ext. į EGB= int. Z GHD,
And L BGH+L GHD = 2 rt. Zs.

PROP. XXIX. THEOR. 30. 1 Ea. Straight lines which are parallel to the same

straight line, are parallel to each other.

PROP. XXX. PROB. 31. 1 Eu. To draw a straight line through a given point,

parallel to a given straight line. Let A be the given point, BC the given str. line. To draw a str. line through A || BC.

In BC take any pt. D,

join AD,
make DAE= L ADC,

Prop. 22.
prod. EA to F,
then shall EF || BC.