greater than the base of the other, the angle_contained by the sides of that 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 to DE, and AC to DF, but the base BO greater than the base EF: the angle BAC shall be greater than the angle

therefore the angle BAC is not equal to the angle EDF. Neither is the angle BAC less than the angle EDF,

for then the base BC would be less than the base EF; [I. 24. but it is not;

[Hypothesis. 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.

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, namely, either the sides adjacent to the equal angles, or sides which are opposite to equal angles in each, 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; and let them have also one side equal to one side; and first let those sides be equal which are adjacent to the equal angles in the two triangles, namely, BC to EF: the other sides shall be equal, each to each, namely, AB to DE, and AC to DF, and the third

angle BAC equal to the third angle EDF.

For if AB be not equal to DE, one of them must be greater than the other. Let AB be the greater, and make BG

equal to DE,

and join GC.

[I. 3.

B

Then because GB is equal to DE,

and BC to EF;

[Construction. [Hypothesis.

the two sides GB, BC are equal to the two sides DE, EF, each to each;

and the angle GBC is equal to the angle DEF; [Hypothesis. therefore the triangle GBC is equal to the triangle DEF, and the other angles to the other angles, each to each, to which the equal sides are opposite;

[I. 4.

therefore the angle GCB is equal to the angle DFE. But the angle DFE is equal to the angle ACB. [Hypothesis.

Therefore the angle GCB is equal to the angle ACB,[Ax. 1. the less to the greater; which is impossible.

Therefore AB is not unequal to DE,

that is, it is equal to it;

and BC is equal to EF;

[Hypothesis.

therefore the two sides AB, BC are equal to the two sides DE, EF, each to each;

and the angle ABC is equal to the angle DEF; [Hypothesis. therefore the base AC is equal to the base DF, and the third angle BAC to the third angle EDF.

[I. 4.

Next, let sides which are opposite to equal angles in each triangle be equal to one another, namely, AB to DE: likewise in this case the other sides shall be equal, each to each, namely, BC to EF, and AC to DF, and also the third angle BAC equal to the third angle EDF. For if BC be not

and AB to DE;

[Construction. [Hypothesis.

the two sides AB, BH are equal to the two sides DE, EF, each to each;

and the angle ABH is equal to the angle DEF ;[Hypothesis, therefore the triangle ABH is equal to the triangle DEF, and the other angles to the other angles, each to each, to which the equal sides are opposite;

[I. 4.

therefore the angle BHA is equal to the angle EFD. But the angle EFD is equal to the angle BCA. [Hypothesis. 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 opposite angle BCA ;

which is impossible.

Therefore BC is not unequal to EF,

[I. 16.

that is, it is equal to it;

and AB is equal to DE;

[Hypothesis.

therefore the two sides AB, BC are equal to the two sides DE, EF, each to each;

and the angle ABC is equal to the angle DEF; [Hypothesis.

therefore the base AC is equal to the base DF, and the third angle BAC to the third angle EDF.

Wherefore, if two triangles &c. Q.E.D.

[I. 4.

PROPOSITION 27. THEOREM.

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

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

For if not, AB and CD, being produced, will meet either towards B, D or towards A, C. Let them be produced and meet towards B, D at the point G.

A E

Therefore GEF is a triangle, and its exterior angle AEF is greater than the interior opposite angle EFG ; [I. 16. But the angle AEF is also equal to the angle EFG; [Hyp. which is impossible.

Therefore AB and CD being produced, do not meet towards B, D.

In the same manner, it may be shewn that they do not meet towards A, C.

But those straight lines which being produced ever so far both ways do not meet, are parallel.

Therefore AB is parallel to CD.

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

[Definition 35.

If a straight line falling on two other straight lines, make the exterior angle equal to the interior and opposite angle on the same side of the line, or make the interior angles on 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 on the two straight lines AB, CD, make the exterior angle EGB equal to the interior and opposite angle GHD on the same side, or make the interior angles on the same side BGH, GHD together equal to two right angles: AB shall be parallel to CD.

Because the angle EGB is equal to the angle GHD,[Hyp. and the angle EGB is also equal A to the angle AGH,

[I. 15. therefore the angle AGH is equal to the angle GHD; [Ax.1. and they are alternate angles; therefore AB is parallel to CD.

C

E

-B

[I. 27.

Again; because the angles BGH, GHD are together equal to two right angles,

[Hypothesis.

and 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.

Takeaway the common angle BGH; therefore the remaining angle AGH is equal to the remaining angle GHD; [Axiom 3. and they are alternate angles;

therefore AB is parallel to CD.

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

[I. 27.

PROPOSITION 29. THEOREM.

If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles.

Let the straight line EF fall on the two parallel straight lines AB, CD: the alternate angles AGH, GHD shall be equal to one another, and the exterior angle EGB shall be equal to the interior and opposite angle