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Book I.

a. 16. 1.

IF

PROP. XXVII. THEOR.

F a straight line falling upon two other straight lines makes the alternate angles equal to one another, these two straight lines fhall be parallel.

Let the ftraight line EF which falls upon the two ftraight 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 BD or towards AC. let them be produced and meet towards BD in the point G; therefore GEF is a triangle, and its exterior angle AEF is greater than the interior and oppofite angle EFG; but it is alfo equal to it, which is impoffible. therefore AB and CD being produced do not meet towards BD. in like manner it may be demonftrated that they do not meet towards AC. but thofe ftraight lines which meet nei

A

C/F

E/

B

D

b. 35. Def ther way tho' produced ever fo far are parallel b to one another. AB therefore is parallel to CD. wherefore if a straight line, &c. Q. E. D,

IF a

PROP. XXVIII. THEOR.

a ftraight line falling upon two other straight lines

makes the exterior angle equal to the interior and oppofite upon the fame fide of the line; or makes the interior angles upon the fame fide together equal to two right angles; the two ftraight lines shall be parallel to one another.

E

Let the ftraight line EF which falls upon the two straight lines AB, CD make the exterior angle EGB equal to the interior and oppofite angle GHD upon the fame fide; or make the interior angles A

on the fame fide BGH, GHD to

gether equal to two right angles, C
AB is parallel to CD.

Because the angle EGB is equal
to the angle GHD, and the angle

G

B

-D

H

F

自然

EGB equal to the angle AGH, the angle AGH is equal to the Book I. angle GHD; and they are the alternate angles; therefore AB is

parallel to CD. again, because the angles BGH, GHD aṛe equal ‘a. 15. 1.

c. By Hyp, d. 13. 1.3

to two right angles, and that AGH, BGH are alfo equal to twob. 27. 1.
right angles; the angles AGH, BGH are equal to the angles BGH,
GHD. take away the common angle BGH, therefore the remain-
ing 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.

See the

Notes on

Fa ftraight line falls upon two parallel ftraight lines,
IF
it makes the alternate angles equal to one another; this Propo-
and the exterior angle equal to the interior and oppofite fition.
upon the fame fide; and likewife the two interior angles
upon the fame fide together equal to two right angles.

Let the ftraight line EF fall upon the parallel straight lines AB,
CD. the alternate angles AGH, GHD are equal to one another;
and the exterior angle EGB is equal to the interior and oppofite,
upon the fame fide, GHD; and the
two interior angles BGH, GHD upon
the fame fide are together equal to
two right angles.

For if AGH be not equal to GHD,
one of them must be greater than the-

E

A

B

H

D

F

other; let AGH be the greater. and be-C
cause the angle AGH is greater than
the angle GHD, add to each of them

a

the angle BGH; therefore the angles AGH, BGH are greater than
the angles BGH, GHD. but the angles AGH, BGH are equal a to a. 13. 1.
two right angles; therefore the angles BGH, GHD are lefs than
two right angles. but those straight lines which with another straight
line falling upon them make the interior angles on the fame fide
lefs than two right angles, do meet together if continually pro- 12. Aș.
duced; therefore the straight lines AB, CD if produced far enough notes on
shall meet. but they never meet, fince they are parallel by the Hy-this Propo-
pothesis. therefore the angle AGH is not unequal to the angle GHD, fition.
that is, it is equal to it. but the angle AGH is equal b to the angle b. 15. 1.
EGB; therefore likewife EGB is equal to GHD. add to each of

See the

c

Book I these the angle BGH, therefore the angles EGB, BGH are equal to the angles BGH, GHD; but EGB, BGH are equal to two right angles; therefore alfo BGH, GHD are equal to two right angles. wherefore if a straight line, &c. Q. E. D.

13.1.

a. 29. I.

b. 27. 1.

ST

PROP. XXX. THEOR.

TRAIGHT lines which are parallel to the fame straight line, are parallel to one another.

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

Let the ftraight line GHK cut AB, EF, CD; and because GHK cuts the parallel straight lines AB,

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nate angles; therefore AB is parallel to CD. wherefore straight lines, &c. Q. E. D.

PROP. XXXI. PRO B.

O draw a straight line thro' a given point parallel to a given straight line.

Let A be the given point, and BC the given straight line; it is

a. 23. I.

required to draw a straight line thro'.

the point A, parallel 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 the angle DAE e-B

D

A F

qual to the angle ADC; and produce the straight line EA to F. Because the straight line AD which meets the two straight lines BC, FF, makes the alternate angles EAD, ADC equal to one b. 27. 1. another, EF is parallel to BC. therefore the ftraight line EAF is

b

drawn thro' the given point A parallel to the given ftraight line Book I. BC. Which was to be done.

IF

PROP. XXXII. THEOR.

Fa fide of any triangle be produced, the exterior angle is equal to the two interior and oppofite 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 fides BC be produced to D. the exterior angle ACD is equal to the two interior and oppofite angles CAB, ABC; and the three interior angles of the triangle, viz. ABC, BCA, CAB are together equal to two

right angles.

Thro' the point C draw CE parallel to the straight line AB. a. 31. 1. and becaufe AB is parallel to CE, and AC meets them, the

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A

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the angle ACE was fhewn to be equal to the angle BAC, therefore the whole exterior angle ACD is equal to the two interior and oppofite angles CAB, ABC. to these equals add the angle ACB, and the angles ACD, ACB are equal to the three angies CBA, BAC, ACB. but the angles ACD, ACB are equal to two right angles; c. 13. I. therefore alfo the angles CBA, BAC, ACB are equal to two right angles. wherefore if a fide of a triangie, &c. Q. E. D.

c

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Book I. angles. And, by the preceding Propofition, all the angles of thefe triangles are equal to twice as many right angles as there are triangles, that is, as there are fides of the figure. and the fame angles are equal to the angles of the figure, together with the angles at a. a. Cor. the point F which is the common Vertex of the triangles; that is, together with four right angles. Therefore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has fides.

15 1.

b. 13. 1.

2. 19. 1.

b. 4. I.

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

Becaufe every interior angle ABC with its adjacent exterior ABD is equal to two right angles; therefore all the interior together with all the exterior angles of the figure, are equal to twice as many right angles as there are fides of the figure, that is, by the foregoing Corollary, they are equal

A

D

B

to all the interior angles of the figure, together with four right angles. therefore all the exterior angles are equal to four right angles.

PROP. XXXIII. THE OR.

THE ftraight lines which join the extremities of two equal and parallel ftraight lines, towards the

fame parts, are alfo themselves equal and parallel.

Let AB, CD be equal and parallel ftraight lines, and joined towards the fame parts by the straight A

lines AC, BD; AC, BD are alfo
equal and parallel.

Join BC, and because AB is pa-
rallel to CD, and BC meets them;
the alternate angles ABC, BCD

B

D

are equal; and because AB is equal to CD, and BC common to the two triangles ABC, DCB, the two fides AB, BC are equal to the two DC, CB; and the angle ABC is equal to the angle BCD; therefore the bafe AC is equal to the bafe BD, and the triangle ABC to the triangle BCD, and the other angles to the other an

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