all the degrees from greater to less, a varying magnitude must evidently encounter the single intermediate limit of equality. Wherefore, there is a certain position CD, in which the line revolving about the point F makes the exterior angle EFC equal to the interior EGA, and at the same instant of time meets AB neither towards the one part nor the other, or is parallel to it.

And now, since EFC is proved to be equal to EGA, and is also equal to the vertical angle GFD; the alternate angles FGA and GFD are equal. Again, because GFD and FGA are equal, add the angle FGB to each, and the two angles GFD and FGB are equal to FGA and FGB; but the angles FGA and FGB, on the same side of AB, are equal to two right angles, and consequently the interior angles GFD and FGB are likewise equal to two right angles.

Cor. Since the position CD is individual, or that only one straight line can be drawn through the point F parallel to AB, it follows that the converse of the proposition is likewise true, and that those three properties of parallel lines are criteria for distinguishing parallels.

PROP. XXIII. PROB.

Through a given point, to draw a straight line parallel to a given straight line.

To draw, through the point C, a straight line parallel to AB.

In AB take any point D, join CD, and at the point C make (I. 4.) an angle DCE equal to CDA ; CE is parallel to AB.

E

D

B

For the angles CDA and DCE, thus formed equal, are the alternate angles which CD makes with the straight lines CE and AB, and, therefore, by the corollary to the last proposition, these lines are parallel.

PROP. XXIV. THEOR.

Parallel lines are equidistant, and equidistant straight lines are parallel.

The perpendiculars EG, FH, let fall from any points E, F in the straight line AB upon its parallel CD, are equal; and if these perpendiculars be equal, the straight lines AB and CD are parallel.

For join EH: and because each of the interior angles EGH and FHG is a right angle, they are together equal to two right angles, and consequently the perpendiculars EG and FH are (I. 22. cor.)

But, EF being parallel to GH,

the alternate angles EHG and HEF are likewise equal; and thus the two triangles HGE and HFE, having the angles HEG and EHG respectively equal to EHF and HEF, and the side EH common to both, are (I. 20.) equal, and hence the side EG is equal to FH.

Again, if the perpendiculars EG and FH be equal, the two triangles EGH and EFH, having the side EG equal to FH, EH common, and the contained angle HEG equal to EHF, are (I. 3.) equal, and therefore the angle EHG equal to HEF, and (I. 22.) the straight line AB parallel to CD.

PROP. XXV. THEOR.

The opposite sides of a rhomboid are parallel.

qua

If the opposite sides AB, DC, and AD, BC of the drilateral figure ABCD be equal, they are also parallel. For draw the diagonal AC. And because AB is equal to DC, BC to AD, and AC is common; the two triangles ABC and ADC are (I. 2.) equal. Consequently the angle ACD is equal D

to CAB, and therefore the side AB (I. 22. cor.) parallel to CD; and, for the same reason, the angle CAD being equal to ACB, the side AD is parallel to BC.

Cor. Hence the angles of a square or rectangle are all of them right angles; for the opposite sides being equal, are parallel; and if the angle at A be right, the other interior one at B is also a right angle (I. 22.), and consequently the angles at C and D, opposite to these, are right. On this proposition depends the construction of the instrument called a Parallel Ruler.

PROP. XXVI. THEOR.

The opposite sides and angles of a parallelogram are equal.

Let the quadrilateral figure ABCD have the sides AB and BC parallel to CD and AD; these are respectively equal, and so are likewise the opposite angles at A and C, and at B and D.

For join AC. Because AB is parallel to CD, the alternate angles BAC and ACD are (I. 22.) equal; and since AD is parallel to BC, the alternate angles ACB and CAD are also equal. Where

D

B

fore the triangles ABC and ADC, having the angles CAB and ACB equal to ACD and CAD, and the interjacent side AC common to both, are (I. 20.) equal. Consequently, the side AB is equal to CD, and the side BC to AD; and these opposite sides being thus equal, the opposite angles (I. 25.) must be likewise equal.

Cor. Hence the diagonal divides a rhomboid or parallelogram into two equal triangles.

PROP. XXVII. THEOR.

If the parallel sides of a trapezoid be equal, the other sides are likewise equal and parallel.

Let the sides AB and DC be equal and parallel; the sides AD and BC are themselves equal and parallel.

For draw the diagonal AC. Because AB is parallel to CD, the alternate angles CAB and ACD are (I. 22.) equal; and the triangles ABC and ADC, having the side AB equal to CD, AC common to both, and the contained angle CAB equal to ACD, are, therefore, equal (I. 3.). Whence the side BC is equal to AD,

D

and the angle ACB equal to CAD; but these angles being alternate, BC must also be parallel to AD (I. 22. cor.)

PROP. XXVIII. THEOR.

Lines parallel to the same straight line, are parallel to each other.

If the straight line AB be parallel to CD, and CD parallel to EF; then is AB parallel to EF.

For let a straight line GH cut these lines.

And because AB is parallel to CD, the exterior angle GIA is equal (I. 22.) to the interior GKC; and since CD is parallel to EF, this angle GKC is, for the same reason, equal to GLE. There

B

K

D

E

LF

C

H

fore the angle GIA is equal to GLE, and consequently AB is parallel to EF (I. 22. cor.)

PROP. XXIX. THEOR.

Straight lines drawn parallel to the sides of an

angle, contain an equal angle.

for the same reason, GAB is equal to GDE; there consequently remains the angle BAC equal to EDF.