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Of the two sides DE, DF, let DE be not greater than DF, at the point D, in the line DE, and on the same side of it as DF, make the angle EDG equal to the angle BAC; (1. 23.) make DG equal to DF or AC, (1. 3.) and join EG, GF. Then, because DE is equal to AB, and DG to AC, the two sides DE, DG are equal to the two AB, AC, each to each, and the angle EDG is equal to the angle BAC; therefore the base EG is equal to the base BC. (1.4.) And because DG is equal to DF in the triangle DFG, therefore the angle DFG is equal to the angle DGF; (1. 5.) but the angle DGF is greater than the angle EGF; (ax. 9.) therefore the angle DFG is also greater than the angle EGF; much more therefore is the angle EFG greater than the angle EGF. And because in the triangle EFG, the angle EFG is greater than the angle EGF,

and that the greater angle is subtended by the greater side; (I. 19.) therefore the side EG is greater than the side EF; but EG was proved equal to BC; therefore BC is greater than EF. Wherefore, if two triangles, &c. Q. E.D.

PROPOSITION XXV. THEOREM.

If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of one greater than the base of the other; the angle contained by the sides of the one which has the greater base, shall be greater than the angle contained by the sides, equal to them, of the other.

Let ABC, DEF be two triangles which have the two sides AB, AC, equal to the two sides DE, DF, each to each, namely, AB equal to DE, and AC to DF; but the base BC greater than the base EF. Then the angle BAC shall be greater than the angle EDF.

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For, if the angle BAC be not greater than the angle EDF,
it must either be equal to it, or less than it.
If the angle BAC were equal to the angle EDF,
then the base BC would be equal to the base EF; (1. 4.)
but it is not equal, (hyp.)

therefore the angle BAC is not equal to the angle EDF.
Again, if the angle BAC were less than the angle EDF,
then the base BC would be less than the base EF; (1. 24.)
but it is not less, (hyp.)

therefore the angle BAC is not less than the angle EDF; and it has been shewn, that the angle BAC is not equal to the angle EDF; therefore the angle BAC is greater than the angle EDF. Wherefore, if two triangles, &c. Q. E. D.

PROPOSITION XXVI. THEOREM.

If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal angles in each, or the sides opposite to them; then shall the other sides be equal, each to each, and also the third angle of the one equal to the third angle of the other.

Let ABC, DEF be two triangles which have the angles ABC, BCA, equal to the angles DEF, EFD, each to each, namely, ABC to DEF, and BCA to EFD; also one side equal to one side.

First, let those sides be equal which are adjacent to the angles that are equal in the two triangles, namely, BC to EF.

Then the other sides shall be equal, each to each, namely, AB to DE, and AC to DF, and the third angle BAC to the third angle EDF.

G

D

B

For, if AB be not equal to DE,
one of them must be greater than the other.
If possible, let AB be greater than DE,
make BG equal to ED, (1. 3.) and join GC.
Then in the two triangles GBC, DEF,

because GB is equal to DE, and BC to EF, (hyp.) the two sides GB, BC are equal to the two DE, EF, each to each; and the angle GBĈ is equal to the angle DEF; therefore the base GC is equal to the base DF, (1. 4.)

and the triangle GBC to the triangle DEF,

and the other angles to the other angles, each to each, to which the equal sides are opposite;

therefore the angle GCB is equal to the angle DFE;

but the angle ACB is, by the hypothesis, equal to the angle DFE; wherefore also the angle GCB is equal to the angle ACB; (ax. 1.) the less angle equal to the greater, which is impossible; therefore AB is not unequal to DE,

that is, AB is equal to DE.

Hence, in the triangles ABC, DEF;

because AB is equal to DE, and BC to EF, (hyp.) and the angle ABC is equal to the angle DEF; (hyp.) therefore the base AC is equal to the base DF, (1. 4.) and the third angle BAC to the third angle EDF.

Secondly, let the sides which are opposite to one of the equal angles in each triangle be equal to one another, namely, AB equal to DE. Then in this case likewise the other sides shall be equal, AC to DF, and BC to EF, and also the third angle BAC to the third angle EDF.

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For if BC be not equal to EF,

one of them must be greater than the other.
If possible, let BC be greater than EF;
make BH equal to EF, (1. 3.) and join AH.
Then in the two triangles ABH, DEF,
because AB is equal to DE, and BH to EF,
and the angle ABH to the angle DEF; (hyp.)
therefore the base AH is equal to the base DF, (1. 4.)
and the triangle ABH to the triangle DEF,

and the other angles to the other angles, each to each, to which the equal sides are opposite;

therefore the angle BHA is equal to the angle EFD;
but the angle EFD is equal to the angle BCA; (hyp.)
therefore the angle BHA is equal to the angle BCA, (ax. 1.)
that is, the exterior angle BHA of the triangle AHC, is
equal to its interior and opposite angle BCA;
which is impossible; (1. 16.)

wherefore BC is not unequal to EF,
that is, BC is equal to EF.

Hence, in the triangles ABC, DEF;

because AB is equal to DE, and BC to EF, (hyp.) and the included angle ABC'is equal to the included angle DEF; (hyp.) therefore the base AC is equal to the base DF, (1. 4.) and the third angle BAC to the third angle EDF.

Wherefore, if two triangles, &c.

Q.E.D.

PROPOSITION XXVII. THEOREM.

If a straight line falling on two other straight lines, make the alternate angles equal to each other; these two straight lines shall be parallel.

Let the straight line EF, which falls upon the two straight lines AB, CD, make the alternate angles AEF, EFD, equal to one another. Then AB shall be parallel to CD.

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then AB and CD being produced will meet, either towards A and C, or towards B and D.

Let AB, CD be produced and meet, if possible, towards B and D, in the point G,

then GEF is a triangle.

C

And because a side GE of the triangle GEF is produced to A, therefore its exterior angle AEF is greater than the interior and opposite angle EFG; (1. 16.)

but the angle AEF is equal to the angle EFG; (hyp.)

therefore the angle AEF is greater than, and equal to, the angle EFG; which is impossible.

Therefore AB, CD being produced, do not meet towards B, D. In like manner, it may be demonstrated, that they do not meet when produced towards A, C.

But those straight lines in the same plane, which meet neither way, though produced ever so far, are parallel to one another; (def. 35.) therefore AB is parallel to CD.

Wherefore, if a straight line, &c. Q.E.D.

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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 make 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 falls upon the two straight lines AB, CD, make the exterior angle EGB equal to the interior and opposite angle GHD, upon the same side of the line EF; or make the two interior angles BGH, GHD on the same side together equal to two right angles.

Then AB shall be parallel to CD.

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Because the angle EGB is equal to the angle GHD, (hyp.) and the angle EGB is equal to the angle AGH, (1. 15.) therefore the angle AGH is equal to the angle GHD; (ax. 1.) and they are alternate angles,

therefore AB is parallel to CD. (1.27.)

Again, because the angles BGH, GHD are together equal to two right angles, (hyp.)

and that the angles AGH, BGH are also together equal to two right angles; (I. 13.)

therefore the angles AGH, BGH are equal to the angles BGH, GHD; (ax. 1.)

take away from these equals, the common angle BGH;

therefore the remaining angle AGH is equal to the remaining angle GHD; (ax. 3.)

and they are alternate angles; therefore AB is parallel to CD. (1. 27.)

Wherefore, if a straight line, &c.

Q.E.D.

PROPOSITION XXIX. THEOREM.

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 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. Then the alternate angles AGH, GHD shall be equal to one another; the exterior angle EGB shall be equal to the interior and opposite angle GHD upon the same side of the line EF;

and the two interior angles BGH, GHD upon the same side of EF shall be together equal to two right angles.

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First. For, if the angle AGH be not equal to the alternate angle GHD, one of them must be greater than the other;

if possible, let AGH be greater than GHD,

then because the angle AGH is greater than the angle GHD, add to each of these unequals the angle BGH;

therefore the angles AGH, BGH are greater than the angles BGH, GHD; (ax. 4.)

but the angles AGH, BGH are equal to two right angles; (1.13.) 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 two interior angles on the same side less than two right angles, will meet together if continually produced; (ax. 12.) therefore the straight lines AB, CD, if produced far enough, will meet towards B, D;

but they never meet, since they are parallel by the hypothesis; therefore the angle AGH is not unequal to the angle GHD, that is, the angle AGH is equal to the alternate angle GHD. Secondly. Because the angle AGH is equal to the angle EGB, (1. 15.) and the angle AGH is equal to the angle GHD,

therefore the exterior angle EGB is equal to the interior and opposite angle GHD, on the same side of the line.

Thirdly. Because the angle EGB is equal to the angle GHD, add to each of them the angle BGH;

therefore the angles EGB, BGH are equal to the angles BGH, GHD; (ax. 2.)

but EGB, BGH are equal to two right angles; (1. 13.) herefore also the two interior angles BGH, GHD on the same side of the line are equal to two right angles. (ax. 1.)

Wherefore, if a straight line, &c.

Q.E.D.

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