PAR 
ft-om G draw G H perpendicular to A B. 
Then since E F and G H are botli perpendi- 
cular 10 A B, and are drawn from F and G 
points in C D, G H equals E F (9). Again, 
since angle G H B or G H E is greater 
than angle GEH (l) E G is greater than 
G H (2). Hence E G is greater than EF. 
Therefore E G is not perpendicular to C D 
(7) ; and in the same manner it may be 
shown that no other line can be drawn 
from the point E perpendicular to C D with- 
out coinciding with E F. Therefore E F is 
perpendicular to C D. 
12. Theorem III. If two straight lines be 
perpendicular to the same straight line, they 
are parallel to each other. 
If A B, ([fig. 4) and C D’be both perpen- 
dicular to E F, then A B is parallel to 
CD. If A B be not parallel to C D, let 
GH passing through the point E, be pa- 
rallel to C D. Then since E F is perpen- 
dicular to C D, it is also perpendicular to 
GH(ll). Hence angle HE F is a right 
angle, and therefore equal to angle B E F, 
the less to 'the greater which is absurd. 
Therefore G H is not parallel to C D ; and 
in the same manner it may be shown that no 
other line passing through E, and not coin- 
ciding with A B, is parallel to C D. There- 
fore A B is parallel to C D. 
13. Cor. Hence it appears, that through 
the same point no more than one line can 
be drawn parallel to the same straight 
line. 
It may be thought necessary to remark, 
that the preceding theorem pre-siipposes 
the admission of a postulate, that through 
any point not in a given straight line, a 
straight line may be drawn parallel to 
that straight line, or that straight line pro- 
duced. 
14. Theorem IV. 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. 
If A B, (fig. 5) be parallel to C D, and 
E F cut them in the points H G, then the 
angle A H G equals the alternate angle 
H G D ; the exterior angle E H B equals 
the interior and opposite angle on the same 
side H G D ; and the two interior angles 
on the same side, B N G and H G D are 
•together equal to two right angles. From H 
draw H K perpendicular to C D, and from 
G draw G I perpendicular to A B. Then 
since H K is perpendicular to C D, it is 
PAR 
also perpendicular to AB (it); conse- 
quently GI is parallel to H K ( 12 ). But 
H I and G K are perpendiculai-s to G I, 
from H and K, points in H K ; therefore (9) 
H I equals G K. Hence in triangles GI H, 
H G K, the side H I equals the side G K, 
G I equals H K (9) and the included angle 
G I H equals the included angle H K G ; 
therefore angle I H G equals angle H G K 
(3) . Again, angle E H B equals A H G 
(4) ; therefore it equals H G D. Lastly, 
B N G and H G D are together equal to 
A H G and B H G together ; and therefore 
(5) are equal together to the sum of two 
right angles. 
15. Theorem V. If a straight line falling 
upon two other straight lines makes the al- 
ternate angles equal to one another, those 
two straight lines will be parallel. 
Let the straight line E F, (fig. 6) which 
falls upon the two straight lines A B, C D, 
make the alternate angles A E F, E F D 
equal to one another, then A B is parallel to 
C D. If not, through E draw G H parallel 
to C D. Then the alternate angle GEF 
equals the alternate angle EFD. But 
A E F equals E F D ; therefore A E F is 
equal to GEF, the less to the greater. 
Hence G H is not parallel to C D j and in 
like manner it may be shown that no other 
line passing through the point E, and not 
coinciding with A B is parallel to C D. 
Therefore A B is parallel to C D. 
16. Cor. If a straight line, falling upon 
two otheV straight lines, makes the exterior 
angle equal to the interior and opposite one 
on the same side of the line ; or makes the 
interior angles on the same side equal to two 
fight angles; the two straight lines shall be 
parallel to one another. 
Parallel planes, are such planes as 
have all the perpendiculars drawn betwixt 
them equal to each other. 
Parallel rays, in optics, are those 
which keep at an equal distance from the 
visible object to the eye, which is sup- 
posed to be infinitely remote from the ob- 
ject. 
Parallel ruler, an instrument consist- 
ing of two wooden, brass, &c. rulers equally 
broad every where ; and so joined together 
by the cross blades as to open to different 
intervals, accede and recede, and yet still 
retain their parallelism. See Pentagraph. 
Parallels, or parallel circles, in geo- 
graphy, called also parallels, or circles of la- 
titude, are lesser circles of the sphere con- 
ceived to be drawn from west to east, 
through all the points of the meridian, com- 
mencing from the equator to which tlvey are 
