Book L.

PROP. XXVII. THEOR.

16. 1.

| 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 the straight line EF, which falls upon the two straight hues 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 shallmeet 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, and its exterior angle AEF is greater a than the interior and opposite angle EFG; but it is also equal

E

B
to it, which is impossible; A
therefore AB and CD be.
ing produced do not ineet
towards B, D. In like man-

C
ner it may be demonstrat-

F

D ed that they do not meet towards A, C; but those. straight lines which meet neither way, though produced ever so far, are parallelb to one another. AB therefore is parallel to CD. Wherefore, if a straight line, &c. Q. E. D.

0 35. def.

а

PROP. XXVIII. THEOR.

.

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 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 E
falls upon the two straight lines

-
AB, CD, make the exterior angle

G
EGB equal to the interior and op- A

B
posite angle GHD upon the same
side; or make the interior angles
on the same side BGH, GHD to-C

D gether equal to two right angles;

H
AB is parallel to CD.
Because the angle EGB is equal

F... to the angle GHD, and the angle

EGB equal to the angle AGH, the angle AGH is equal to the Book I. angle GHD; and they are alternate angles; therefore AB is parallel b to CD. Again, because the angles BGH, GHD are a 15. 1, i equal to two right angles; and that AGH, BGH, are also b 27. 1. equal d to two right angles; the angles AGH, BGH are equal By byp.

d 13. 1. to the angles BGH, GHD:take away the coinmon angl. 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. ED.

PROP. XXIX. THEOR.

IF a straight line fall upon two parallel straight See the lines, it makes the alternate angles equal to one ano.

notes on this

proposition. ther; 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.

AD

Let the straight line Er fall upon the parallel straight lines AB, CD; the ahernate angles, AGH, GHD are equal to one another; and the exterior angle EGB is equal to the interior and opposite, upon the same

E
side GHD; and the two inte-
rior angles BGH, GHD upon

H
А

B
the same side are together
equal to two right angles.

For if AGH be not equal to GHD, one of them must be C greater than the other; let AGH be the greater; and because the angle AGH is great. er 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 a to two a 13. 1. 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 12 ax, the same side less than two right angles, do meet* togetier if See the continually produced; therefore the straight lines AB, CD, if notes on

this propoproduced far enough, shall meet; but they never meet, since

sition, they are parallel by the hypothesis; therefore the angle AGH is not unequal to the angle GHD, that is, it is equal to it; but

It b.,15. 1 the angle AGH is equal to the angle EGB; therefore like

Book I.

'c 13. 1.

wise EGB is equal to GHD; add to each of those the angle
BGH; therefore the angles EGB, BGH are equal to the angles
BGH, GHD; but EGB, BGH are equal c to two right angles;
therefore also BGH, GHD are equal to two right angles.
Wherefore, if a straight line, &c. Q. E. D.

PROP. XXX. THEOR.

STRAIGHT lines which are parallel to the same straigin line are parallel to one another.

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; and because

i
GHK cuts the parallel straight
lines AB, EF, the angle AGH is ;
equal a to the angle GHF. Again,

A

B
because the straight line GK cuts
the parallel straight lines EF, CD,

H
the angle GHF is equal to a the

E angle GKD; and it was shown that

K
the angle AGK is equal to the an- C

D
gle GHF; therefore also AĞK is
equal to GKD; and they are alter-
nate angles; therefore AB is paral-
lel b to CD, Wherefore straight lines, &c. Q. E. D.

a 29. 1.

b 27.1.

PROP. XXXI. PROB.

TO draw a straight line through a given point parallel to a given straight line.

23. .

Let A be the given point, and BC the given straight line;
it is required to draw a straight
line through the point A, parallel E

А F
to the straight line BC.

In BC take any point D, and
join AD; and at the point A, in the
straight line AD make a the angle
DAE equal to the angle ADC; B D

с
and produce the straight line EA
to F.

Because the straight line AD, which meets the two straight
lines BC, EF, makes the alternate angles EAD, ADC equal to
otte another, EF is parallel b to BC. Therefore the straight line

EAF is drawn through the given point A parallel to the given Book I. Straight line BC. Which was to be done.

PROP. XXXII. THEOR.

E, .

IF a side of any triangle be produced, the exterior

a angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are equal to two right angles.

Let ABC be a triangle, and let one of its sides BC be produced to D; the exterior angle ACD is equal to the two interior and opposite angles CAB, ABC; and the three interior angles of the triangle, viz. ABC, BCA, CAB, are together equal to two tight angles. Through the point C

A draw CE parallel to the

a 31. 1. straight line AB; and because AB is parallel to CE and AC meets them, the alternate angles BAC, ACE are equalb. Again;

B

C

Do . because AB is parallel to CE, and BD falls upon them, the exterior angle ECD is equal to the interior and opposite angle ABC; but the angle ACE was shown to be equal to the angle BAC; therefore the whole exterior anyle ACD is equal to the two interior and opposite angles CAB, ABC; to these equals add the angle ACB, and the angles ACD, ACB are equal to the three angles CBA, BAC, ACB; but the angles. ICD, ACB are equal to two right angles: therefore also the angles c 13. 1. CBA, BAC, ACB are equal to two right angles. Wherefore, if a side of a triangle, &c. Q. E. D.

Cor. 1. All the interior angles of any rectilingual figure, together D with four right angles, are equal to twice as many right angles as E the figure has sides.

For any rectilineal figure ABCDE can be divided into as many triangles as the figure has sides, by drawing straight lines from a point F within the figure to each of its A

Bangles. And, by the preceding

Book I. proposition, all the angles of these triangles are equal to

twice as many right angles as there are triangles, that is, as.
there are sides of the figure; and the same angies are equal

to the angles of the figure, together with the angles at the a 2 Cor.

point F, which is the common vertex of the triangles: that isa, 15. 1.

together with four right angles. Therefore all the angics of
the figure, together with four right angies are equal to twice
as many right angles as the figure has sides.

COR: 2. All the exterior angles of any rectilineal figure, are
together equal to four right angles.

Because every interior angle

ABC, with its adjacent exterior 13. 1.

ABD, is equalb to two right an.
gles; therefore all the interior,
together with all the exterior
angles of the figure, are equalto
twice as many right angles as
there are sides of the figure;D
that is, by the foregoing corol-
lary, they are equal to all the
interior angles of the figure,. 10-
gether with four right angles; therefore all the exterior an-
gles are equal to four right angles.

А

PROP. XXXIII. THEOR.

THE straight lines which join the extremities of two equal and parallel straight lines, towards the same parts, are also themselves equal and parallel.

Let AB, CD be equal and pa- 4

B
rallel straight lines, and joined to-
wards the same parts by the straight
lines AC, BD; AC, BD are also
equal and parallel.

Join BC; and because AB is
parallel to CD, and BC meets them, C
the alternate angles ABC, BCD are equala; and because AB
is equal to CD, and BC common to the two triangles ABC,
DCB, the two sides AB, BC are equal to the two DC, CB;
and the angle ABC is equal to the angle BCD; therefore the
base AC is equalb to the base BD, and the triangle ABC to the
triangle BCD, and the other angles to the other angles, each
to each, to which the equal sides are opposite: therefore the

2 29.1.