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Book I. but the angle AGH is equalb to the angle EGB; therefore

likewise EGB is equal to GHD; add to each of these the • 15. 1. angle BGH; therefore the angles EGB, BGH are equal 13. 1. 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, &c. Q.E. D.

.PROP. XXX. THEOR. STRAIGHT lines which are parallel to the same straight line are parallel to each other.

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 be

cause GHK cuts the parallel ; straight lines AB, EF, the an* 29. 1. gle AGH is equals to the an

gle GHF. Again, because the “*
straight line GK cuts the pa-
rallel straight lines EF, CD, L-
the angle GHF is equala to .
the angle GKD; and it was
shown that the angle AGK is
equal to the angle GHF;

therefore also AGK is equal to GKD; and they are alter• 27. ). nate angles; therefore AB is parallelb to CD. Wherefore

straight lines, &c. Q. E.D.

PROP. XXXI. PROB. To draw a straight line through a given point parallel to a given straight line..

Let A be the given point, and BC the given straight line; it is required to draw a straight F _ A_ f line through the point A, paral- Jel to the straight line BC.

In BC take any point D, and

join AD; and at the point A, in B D * *3. 1. the straight line AD, makea the

angle DAE equal to the angle ADC; 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,

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ADC equal to one another, EF is parallelb to BC. There- Book I. fore the straight line EAF is drawn through the given point A parallel to the given 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 to

gether equal to two right angles. · Through the point C

draw CE parallela to the
straight line AB; and
because AB is parallel
to CE, and AC meets
them, the alternate an-
gles BAC, ACE, are s

D equalb. Again, because

b29. 1. 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 : c 13, 1. therefore also the angles 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 rectilineal figure, together with four right angles, are equal to E twice as many right angles as 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

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15. It

Book I. each of its angles. And, by the preceding proposition, all w 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 F, which • 2 Cor. is the common vertex of the triangles : that isa, 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.

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 exte- 13. 1. rior ABD, is equalb 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 D
of the figure; that is, by the
foregoing corollary, they are equal 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. 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 parallel straight lines, and join- A ed 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 them, the alternate ana 29. 1. gles ABC, BCD are equala; and because AB is equal to

ČD, and BC common to the two triangles ABC,DČB, 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 * 4. 1. AČ is equalb to the base BD, and the triangle ABC to the

triangle BCD, and the other angles to the other anglesb, each to each, to which the equal sides are opposite; therefore the angle ACB is equal to the angle CBD; and be- Book I. cause the straight 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 e 27. 1. be equal to it. Therefore, straight 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 biserts it.

Because AB is parallel to CD, and BC meets them, the alternate angles ABC, BCD are equal to one anothera; and

a 29. 1. because AC is parallel to BD, and BC meets them, the alterpate angles ACB, CBD, are equala 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 i 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 otherb, viz, the side AB to the side CD, and b 26. 1. AČ 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 equal to the angle BCD; therefore the triangle

Book I. each of its angles. And, by 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 F, which • 2 Cor. is the common vertex of the triangles : that isa, together 15. I 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.

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 exteb 13. 1. rior ABD, is equalb 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 D B
of the figure; that is, by the
foregoing corollary, they are equal 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. THEOR. THE straight lines wliich 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 parallel straight lines, and join- A

AB ed 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 them, the alternate an* 29. 1. gles 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 4. 1. 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; there

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