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IF

PROPOSITION XXV.

THEOREM XVI.

F two triangles (BAC, EDF,) have two fides of the one equal to two fides of the other, each to each, but the bafe (BC) of the one greater than the bafe (EF) of the other; the angle (BAC) oppofite to the greater base (BC), will be alfo greater than the angle (D) oppofite to the leffer bafe (EF). Hypothefis.

I. ABDE.

11. ACDF.

III. BC > EF.

If not,

Thefis. The angle A oppofite to the greater bafe BC, is > Doppofite to the leffor bafe EF.

DEMONSTRATION.

The angle A is either equal or lefs than the angle D.

BECAUSE

CASE I. Suppofe VA to be to V D.

=

ECAUSE VA is to VD (Sup. 1.), & the fides AB, AC, & DE, DF, which contain those V, are equal each to each, (Hyp. 1 & 2.).

1. The base BC is to the base EF.

But the base BC is not to the base EF (Hyp. 3.).

2. Therefore ▾ A cannot be = to V D.

BECAUSE

CASE II. Suppofe VA to be <VD.

ECAUSE VA is <VD (Sup. 2.), & the fides AB, AC, & DE, DF, which contain thofe V are equal, each to each, (Hyp. 1 & 2.).

1. The bafe BC is the base EF.

But the base BC is not the base EF (Hyp. 3.).

2. Therefore A is not <VD.

But it has been fhewn that neither is it equal to it (Cafe. 1.).

3. Confequently, VA, which is oppofite to the greater bafe BC,

is>VD, which is oppofite to the leffer bafe EF.

C. N.

P. 4. B. 1.

P. 24. B. 1.

Which was to be demonftrated,

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IF

PROPOSITION XXVI. THEOREM XVII.

F two triangles (ACB, DFE,) have two angles (A & B) of one, equal to two angles (D & FED) of the other, each to each, & one fide equal to one fide, viz. either the fides, as (AB & DE) adjacent to the equal angles; or the fides, as (AC & DF) oppofite to equal angles in each: then shall the two other fides (AC, BC, or AB, BC,) be equal to the two other fides (DF, EF, or DE, EF,) each to each, & the third angle (C) equal to the third angle (F).

Hypothefis.

I. VAV D.
II. V BV FED.

III. ABDE.

If not,

CASE I

When the equal fides AB, DE, are
adjacent to the equal angles A &D,
B & FED (Fig. 1 & 2.).
DEMONSTRATION.

Thefis.

I. AC DF. II. BC EF. III. VC=VF.

The fides are unequal, & one, as DF will be> the other AC.

Preparation.

=

1. Cut off from the greater fide DF a part DG to AC.

P. 3.

B. 1.

2. From the point G to the point E, draw the straight line GE. Pos. 1.

BECAUSE in the AACB, DGE, the fide AC is to the fide DG,

(Prep. 1.), ABDE (Hyp. 3.), & VA is to V D. (Hyp. 1.).

1. The VB & GED oppofite to the equal fides AC & DG are equal. P. 4. B. 1.

But VB being to V GED (Arg. 1,), & this fame VB being alfo

= to V FED (Hyp. 2.).

2. It follows, that V GED is to V FED.

But FED being the whole & \ GED its part:

3. The whole would be to its part.

4. Which is impoffibe.

5. The fides AC, DF, are therefore not unequal.. 6. Confequently, they are equal, or ACDF.

Which was to be demonftrated. I. Since then in the AACB, DFE, the fide AC is to the fide DF, (Arg. 6.), AB DE (Hyp. 3.), & VA is to VD (Hyp. 1). -. The third fide BC is alfo to the third fide EF, & the VC & F, oppofite to the equal fides AB, DE, are alfo to one another.

Which was to be demonftrated. II & III.

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C. N.

P. 4. B. I.

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The fides AB, DE, are unequal; and one, as DE, will be > the
other AB.

Preparation.

1. Cut off from the greater fide DE, a part DG = to AB.

2. From the point G to the point F, draw the straight line GF.

BECAUS

ECAUSE then in the AACB, DFG, the fide AC is the fide DF (Hyp. 3.), AB = DG (Prep. 1.), & VA is (Hyp. 1.).

P. 3. B. 1.
Pof. 1.

to

to V D,

1. The other V B & DGF, to which the equal fides AC, DF, are oppofite, are to one another.

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2. It follows, that VE is to V DGF.

The angle B being therefore DGF (Arg. 1.), & this fame V B being alfo to VE (Hyp. 2.).

Ax. 1.

But DGF is an exterior V of A GFE, & VE, is its interior oppofite one.

3. Therefore the exterior will be equal to its interior oppofite one. 4. Which is impoffible.

5. Confequently, the fides AB, DE, are not unequal. 6. They are therefore equal, or AB = DE.

Which was to be demonftrated. I. Since then in the AACB, DFE, the fide AC is to the fide DF, (Hyp. 3.), AB=DE (Arg. 6.), & VA is to VD (Ep. 1.). 7. It is evident, that the third fide BC is to the third fide EF, & the VC & F, to which the equal fides AB, DE, are oppofite, are equal to one another.

Which was to be demonftrated. II. & III.
F

P. 16. B. 1.

C. N.

P. 4. B. 1.

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PROPOSITION XXVII. THEOREM XVIII.

IF a ftraight line. (EF), falling upon two other straight lines (AB, CD,) fituated in the fame plane, makes the alternate angles (m & p, or n & 0,) cqual to one another thefe two ftraight lines (AB, CD,) fhall be parallel.

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I. AB, CD, are two straight lines in the fame plane.
II. The line EF cuts them fo that \m\p, or n=vo.

Thefis.
The lines AB, CD,

are plle.

If not,

DEMONSTRATION.

The ftraight lines AB, CD, produced will meet either towards
BD or towards AC.

Preparation.

Let them be produced & meet towards BD in the point M.

BECAUSE

USE then is an exterior angle of A GMH, & Vo its

interior oppofite one.

1. The Vn is > Vo.

But Vn is to Vo (Hyp. 2.).

2. This Vn is therefore not > Vo.

D. 35. B. 1.

Pof. 2.

P. 16. B. I.

C. N.

Which was to be demonstrated.

D. 35. B. 1.

3. Confequently, it is impoffible that the ftraight lines AB, CD, should meet in a point as M.

4. From whence it follows that they are plle ftraight lines.

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I

PROPOSITION XXVIII. THEOREM XIX.

Fa ftraight line (EF) falling upon two other fraight lines (AB, CD,) fituated in the fame plane, makes the exterior angle (m) equal to the interior & oppofite (n) upon the fame fide, or makes the interior angles (a + n) upon the fame fide equal to two right angles; thofe two straight lines AB, CD, fhall be parallel to one another.

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1. They are

to one another.

=

The being therefore to V m (Arg. 1.), & Vn being to the famem (Hyp.).

2. It is evident that Vp is also to V n.

But the equal Vp & n (Arg. 2.), are alfo alternate V.

3. Confequently, the ftraight lines AB, CD, are plle.

Hypothefis.

CASE II.

DEMONSTRATION.

The Yo+nare to 2 L.

Thefis.

P. 15. B. 1.

Ax. 1.

P. 27. B. 1.

AB, CD, are plle, lines,

BECAUSE the ftraight line EF falling upon the straight line AB,

forms with it the adjacent Vo & p.

1. Those Vo+pare to two L.

The Vo+p being therefore to two L. (Arg. 1.), & the Vo+n being alfo to two L (Hyp.).

2. It follows, that the Vo+pare to Yo+n.

And if the common angle o be taken away from both fides.

3. The remaining p & n will be equal to one another.
But thofe equal Vp & n (Arg. 3.), are at the fame time alternate V.

4. Confequently, the ftraight lines AB, CD, are plle.

Which was to be demonftrated.

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