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c

EGB equal a to the angle AGH, the angle AGH is equal Book I. to the angle GHD; and they are the alternate angles; therefore AB is parallel to CD. Again, because the angles BGH, GHD a 15. 1. are equal to two right angles; and that AGH, BGH are also b 27. I. equal to two right angles; the angles AGH, BGH are equal c By hyp. to the angles BGH, GHD: take away the common angle d 13. 1. 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. E. D.

PROP. XXIX. THEOR.

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IF a straight line fall upon two parallel straight See the lines, it makes the alternate angles equal to one ano- notes on ther; and the exterior angle equal to the interior and position. opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles.

Let the straight 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 opposite, upon the same side GHD; and the two interior angles BGH, GHD upon the same side are together equal to two right angles.

For, if AGH be not equal to GHD, one of them must be greater than the other; let AGH be the greater; and because the angle AGH is greater than the an

E

A

G

C.

B

D

H

F

gle 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 right angles; therefore a 13. 1. 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 the same side less than two right angles, do meet together if continually produced; therefore 12. ax. the straight lines AB, CD, if produced far enough, shall meet ; See the but they never meet, since 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 the angle AGH is equal to the angle EGB; therefore likewise EGB is equal to GHD; add to each

b

notes on this proposition. b 15. 1.

Book I. of 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 also BGH, GHD are equal to two right angles. Wherefore, if a straight line, &c. Q. E. D.

c 13. 1.

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

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

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A

B

H

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C

D

Let the straight line GHK cut AB, EF, CD; and because GHK cuts the parallel straight lines AB, EF, the angle AGH is a 29. 1. equal a to the angle GHF. Again, because the straight line GK cuts the parallel straight lines EF, CD, the angle GHF is equal to the angle GKD a; and it was shown that the angle AGK is equal to the angle GHF; therefore also AGK is equal to GKD; and they are alternate angles;

E.

b 27.1. therefore AB is parallel b to CD. Wherefore, straight lines, &c.

Q. E. D.

PROP. XXXI. PROB.

TO draw a straight line through a given point pa

rallel to a given straight line.

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 to the E
straight line BC.

D

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C

In BC take any point D, and join AD; and at the point A, in the a 23. 1. straight line AD, make a the angle DAE equal to the angle ADC; and B 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 one another, EF is parallel b to BC. Therefore the straight line

b 27. 1.

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.

IF a side of any triangle be produced, the exterior 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 right angles.

a

Through the point C draw CE parallel to the straight line AB; and because AB is parallel to CE, and AC meets them, the alternate angles BAC, ACE are equalb. Again, B

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b 29. 1.

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 angle 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 ACD, ACB are equal to two right angles: therefore also the angles CBA, c 13. 1. BAC, ACB are equal to two right angles. Wherefore, if a side of a triangle, &c. Q. E. D.

c

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Book I. the preceding 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 angles are equal to the angles of the figure, together with the angles at the point a 2. Cor. F, which is the common vertex of the triangles; that is a, 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 sides.

15. 1.

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 b 13. 1. 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 sides of the figure : D
that is, by the foregoing corol-
lary, they are equal to all the
interior angles of the figure, to-

A

B

gether with four right angles; therefore all the exterior angles are equal to four right angles.

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- A rallel straight lines, and joined towards 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

C

B

D

a 29. 1. them, the alternate angles ABC, BCD are equal a; 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 equal 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

b 4. 1.

angle ACB is equal to the angle CBD; and because the straight Book I line BC meets the two straight lines AC, BD, and makes the alternate angles ACB, CBD equal to one another, AC is parallel

to BD; and it was shown to be equal to it. Therefore, straight © 27. 1. lines, &c. Q. E. D.

PROP. XXXIV. THEOR.

THE opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them, that is, divides them into two equal parts.

N. B. A parallelogram is a four sided figure, of which the opposite sides are parallel; and the diameter is the straight line joining two of its opposite angles.

Let ACDB be a parallelogram, of which BC is a diameter; the opposite sides and angles of the figure are equal to one another; and the diameter BC bisects it.

Because AB is parallel to CD, A

C

B

D

a 29. 1.

and BC meets them, the alternate angles ABC, BCD are equal a to one another; and because AC is parallel to BD, and BC meets them, the alternate angles ACB, CBD are equal a to one another; wherefore the two triangles ABC, CBD have two angles ABC, BCA in one, equal to two angles BCD, CBD in the other, each to each, and one side BC common to the two triangles, which is adjacent to their equal angles; therefore their other sides shall be equal, each to each, and the third angle of the one to the third angle of the other b, viz. the b 26. 1. side AB to the side CD, and AC to BD, and the angle BAC equal to the angle BDC;, and because the angle ABC is equal to the angle BCD, and the angle CBD to the angle ACB, the whole angle ABD is equal to the whole angle ACD: and the angle BAC has been shown to be equal to the angle BDC; therefore the opposite sides and angles of parallelograms are equal to one another; also, their diameter bisects them; for AB being equal to CD, and BC common, the two AB, BC are equal to the two DC, CB, each to each; and the angle ABC is

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