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PROP. XXVI. THEOR. 27. 1 Eu.
straight lines, make the alternate angles
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.
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,
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,
PROP. XXIX. THEOR. 30. 1 Ea. Straight lines which are parallel to the same
straight line, are parallel to each other.