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DBE. Therefore the arm B F coincides with BE, and ABE is a straight line.

VII.-EUCLID I. 15.

The vertical or opposite angles, made by two straight lines cutting each other, are equal to each other.

Let the two straight lines A B, CD (Fig. 4), cut one another in the point E, the angle A EC shall be equal to the angle DE B, and CEB to AED.

The angles A E C and CE B, on one side of the straight line AB, are together equal to two right angles; and the angles CE B, BED, on one side of the straight line CD, are also equal to two right angles; wherefore the angles AEC, CEB, are together equal to the two CEB, BED. Take away the common angle CE B, and the remaining angle CE A is equal to the remaining angle D E B. In the same way it may be proved that the angles CE B, A E D, are equal.

Cor.-The four angles made by two straight lines crossing each other at a right angle, are all right angles.

VIII.

The perpendicular is transverse to the straight line on which it stands, and conversely a straight line meeting another in a transverse direction is at right angles to it.

Let CD (Fig. 5), be a straight line perpendicular to CA, and let CA be produced on the other side of C, to a point B, and let CA point to the left, CB to the right. Now let the point C remain fixed while the plane CADB is turned over upon its face, so that the position of every point in the system shall be precisely reversed with respect to right and left of the point C, while no difference is made in the position of points in a direction up and down the paper; and let A'B'D' be the position of A, B, D in the reversed system. Then as CB formerly pointed directly to the right, it will now point left, and thus, as C B', will coincide with C A, and CA' for the same reason with CB. But the angle BCD equals the angle ACD, being both right angles; therefore the angle B'CD' (which is the angle B C D in its new position) or A C D' is equal to the angle AC D. Therefore CD' coincides with CD, or, in other words, there is no change in the position of CD. No point, therefore, in CD, can lie either to the right or left of the point C, or the spectator, advancing along CD, would be without motion in the direction C A or CB; that is to say, CD is transverse to C A and C B.

Next, let CD be transverse to CA, CB. Then let the position of the plane CADB be reversed as before, and A', B', be the new positions of the points A, B.

Then, because CD is transverse to C B, no point

in CD lies either to the right or left of C, and the position of CD will remain unaltered in the reversed figure. Moreover, CB', for the same reason as before, will coincide with CA, and the angle B'CD with the angle AC D; therefore, the angle BCD is equal to A CD, or they are both right angles, and CD is perpendicular to CA or C B.

Cor. If one direction be transverse to a second, the second is also transverse to the first.

IX.

If two straight lines pointing in the same direction make equal angles with two other straight lines in the same plane, the latter are also in the same direction.

And conversely if two straight lines in the same direction with each other intersect two other straight lines also in the same direction with each other, the angles between each pair of intersecting lines are equal.

If A B, DE (Fig. 6) be straight lines in the same direction and they meet the lines AC, DF in the same plane, making the angle BAC equal to E D F, the straight line A C shall be in the same direction with D F.

And conversely, if A B and AC be in the same direction with DE, DF respectively, the angle BAC shall be equal to the angle ED F.

First, let the angle B A C be equal to the angle

EDF; join AD, and let the angle B A C slide along the line AD without any change of direction in any of the lines of the system until the point A is brought to coincide with D. Then, because DE is in the same direction with AB, the two lines will coincide; and because the angle BAC is equal to EDF, when A is brought to D the angles will also coincide, and the line A C will coincide with DF; therefore AC is in the same direction with D F.

In the next place, let A B, A C be in the same directions respectively with DE, DF; and the same construction being made when A is brought to D, A B will coincide with DE, and AE with DF; therefore the angle BAC is equal to the angle ED F.

X.

Two parallel lines may always be included in the same plane.

Let A B, CD (Fig. 7) be parallel straight lines, AC a straight line cutting them, and let A E, C F be straight lines in the direction of the normal to the plane A C, A B.

Then, because CD is in same direction with A B, it will be transverse to every direction to which A B is transverse, and, therefore, to the direction AE or CF. CD is, therefore, a line in the same plane with A B and A C.

XI.

Parallel straight lines make equal angles with a straight line cutting them; and, conversely, straight lines in the same plane making equal angles with a straight line cutting them are parallel to each other.

Let A B, C D (Fig. 7) be parallel straight lines, ACG a straight line cutting them in A and C respectively; the angle GA B shall be equal to G C D.

Because ACG is a straight line, A C and CG are in the same direction; and A B and C D, being parallel, are also in the same direction with each other. Wherefore (Prop. 9) the angle CAB is equal to GCD.

Next, let AB, CD, be straight lines in the same plane, making the angle CAB equal to GCD; CD shall be parallel to A B.

Because AC is in the same direction with CG, and the angle CAB is equal to G C D, therefore (Prop. 9, case 2) CD is in the same direction with A B, that is, CD and A B are parallel.

XII.

Parallel straight lines do not meet if produced ever so far both ways.

Let any straight lines A E, CE (Fig. 4) meet in the point E, and let them be produced beyond E to B and D; they cannot again meet if produced ever so far (Prop. 2). Then CE and AE are in the directions ED and E B respectively,

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