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The fide AC, therefore, cannot be greater than the fide DF; and, in the fame manner, it may be fhewn that it cannot be less; confequently it must be equal to it.

And, fince the two fides AC, AB, are equal to the two fides DF, DE, each to each, and the angle CAB is equal to the angle FDE, the fide BC will also be equal to the fide EF, and the two triangles will be equal in all refpects (Prop. 4.) Q. E. D.

PROP. XXII. THEOREM.

If a right line interfect two other right lines, and make the alternate angles equal to each other, thofe lines will be parallel.

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Let the right line EF interfect the two right lines AB, CD, and make the alternate angles AEF, EFD equal to each other; then will AB be parallel to CD.

For, if they be not parallel, let them be produced, and they will meet each other, either on the fide AC, or on the fide BD (Def. 20.)

Suppose them to meet in the point G, on the fide BD. Then, fince FGE is a triangle, the outward angle AEF is greater than the inward oppofite angle EFD (Prop. 16.)

But the angles, AEF, EFD, are equal to each other (bag Hyp.); whence they are equal and unequal at the fame time, which is abfurd.

The lines AB, CD, therefore, cannot meet on the fide BD; and, in the same manner, it may be fhewn that they cannot meet on the fide AC; consequently they must be parallel to each other (Def. 20.) QE.D.

COROLL. Right lines which are perpendicular to the fame right line are parallel to each other.

PROP. XXIII. THEOREM.

;

If a right line interfect two other right lines, and make the outward angle equal to the inward oppofite one, on the fame fide or the two inward angles, on the fame fide, together equal to two right angles, those lines will be parallel.

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Let the right line EF interfect the two right lines AB, CD, and make the outward angle EGB equal to the inward angle GHD; or the two inward angles BGH, GHD together equal to two right angles; then will AB be parallel to CD.

For, fince the angles EGB, GHD are equal to each other (by Hyp.), and the angles AGH, EGB are alfo equal to each other (Prop. 15.), the angle AGH will be equal to the angle GHD (Ax. 1.)

But when a right line interfects two other right lines, and makes the alternate angles equal to each other, those lines will be parallel (Prop. 22.); therefore AB is parallel

to CD.

Again, fince the angles BGH, GHD are, together, equal to two right angles (by Hyp.), and AGH, BGH are, also, equal to two right angles (Prop. 13.), the angles AGH, BGH will be equal to the angles BGH, GHD (Ax. 1.)

And, if the common angle BGH be taken away, the remaining angle AGH will be equal to the remaining angle GHD (Ax. 3.)

But these are alternate angles; therefore, in this cafe, AB will, also, be parallel to CD (Prop. 22.) Q. E. D.

PROP. XXIV. THEOREM.

If a right line interfect two parallel right lines, it will make the alternate angles equal to each other.

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Let the right line EF interfect the two parallel right lines AB, CD; then will the angle AEF be equal to the alternate angle EFD.

For

For if they be not equal, one of them must be greater than the other ; let EFD be the greater; and make the angle EFB equal to AEF (Prop. 20.)

Then, fince AB, CD are parallel, the right line FB, which interfects CD, being produced, will meet AB in fome point B (Pof. 4.)

And, fince EFB is a triangle, the outward angle AEF will be greater than the inward oppofite angle EFB (Prop. 16.)

But the angles AEF, EFB are equal to each other (by Conft.) whence they are equal and unequal, at the same time, which is abfurd.

The angle EFD, therefore, is not greater than the angle AEF; and, in the fame manner, it may be thewn that it is not lefs; confequently they must be equal to each other. QE. D.

COROLL. Right lines which are perpendicular to one of two parallel right lines, are alfo perpendicular to the other.

PROP. XXV. THEOREM.

If a right line interfect two parallel right lines, the outward angle will be equal to the inward oppofite one, on the fame side; and the two inward angles, on the fame fide, will be equal to two right angles.

H

G

E

Let the right line EF interfect the two parallel right lines AB, CD; then will the outward angle EGB be equal to the inward oppofite angle GHD; and the two inward angles BGH, GHD will be equal to two right angles.

For, fince the right line EF interfects the two parallel right lines AB, CD, the angle AGH will be equal to the alternate angle GHD (Prop. 24.)

But the angle AGH is equal to the oppofite angle EGB (Prop. 15.); therefore the angle EGB will, alfo, be equal to the angle GHD.

Again, fince the right line BG falls upon the right line EF, the angles EGB, BGH, taken together, are equal. to two right angles (Prop. 13.)

taken to

But the angle EGB has been fhewn to be equal to the angle GHD; therefore, the angles BGH, GHD, gether, will, alfo, be equal to two right angles.

D

Q. E. D.

COROLL.

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