SECTION III.

PARALLELS AND PARALLELOGRAMS.

Def. 35. Parallel straight lines are such as are in the same plane and being produced to any length both ways do

not meet.

Axiom 5. Two straight lines that intersect one another cannot both be parallel to the same straight line.

Def. 36. A trapezium is a quadrilateral that has only one pair of opposite sides parallel.

This figure is sometimes called a trapezoid.

Def. 37. A parallelogram is a quadrilateral whose opposite sides are parallel.

Def. 38. When a straight line intersects two other straight lines it makes with them eight angles, which have received special names in relation to one another.

Thus in the figure 1, 2, 7, 8 are called exterior angles, and 3, 4, 5, 6, interior angles; again, 4 and 6, 3 and 5, are called alternate angles; lastly, I and 5, 2 and 6, 3 and 7, 4 and 8, are called corresponding angles.

2/1

8/4

6/5

18

Def. 39. The orthogonal projection of one straight line on another straight line is the portion of the latter intercepted

between perpendiculars let fall on it from the extremities of the former.

Thus the projections of AB, CD on EF are the lines ab, cd respectively.

E

It is clear that the line EF must be supposed indefinitely long. There could be no projection of AB on the terminated line GF.

THEOREM 21.

If one straight line intersects two other straight lines so as to make the alternate angles equal, the straight lines are parallel.

Part. En. Let ABCD intersect EF and GH, and make the angle EBC equal to its alternate angle BCH; it is required to prove that EF is parallel to GH.

Proof. For if EF and GH meet towards F, H, they would form a triangle with BC;

and EBC would be its exterior angle, and therefore greater than the interior and opposite angle BCH.

But EBC is equal to BCH,

(Th. 9.)

(Hyp.)

(Def. 35.) Q. E. D.

therefore EF and GH do not meet towards F, H.

Similarly they do not meet towards E, G;

that is, EF is parallel to GF*.

THEOREM 22.

If two straight lines are parallel, and are intersected by a third straight line, the alternate angles are equal.

Part. En. Let EF and GH be parallel straight lines, and let ABCD intersect them;

it is required to prove that the alternate angles EBC, BCH are equal.

Proof. For if EBC were not equal to BCH,

let some other line LBM be drawn through B making the

angle LBC equal to the alternate angle BCH;

then LM would be parallel to GH.

But EF is parallel to GH;

(Th. 21.)

(Hyp.)

that is, two intersecting lines LM, EF would be both parallel to GH; which is impossible.

(Ax. 5.)

Therefore EBC is equal to BCH, that is, the alternate

angles are equal.

* Euclid, I. 27.

+ Euclid, I. 29.

Q. E. D.

THEOREM 23.

If a straight line intersects two other straight lines and makes either a pair of alternate angles equal, or a pair of corresponding angles equal, or a pair of interior angles on the same side supplementary; then, in each case, the two pairs of alternate angles are equal, and the four pairs of corresponding angles are equal, and the two pairs of interior angles on the same side are supplementary.

Part. En. Let the straight line ABCD intersect the two straight lines EF, GH, and make the alternate angles EBC, BCH equal; then will the other alternate angles FBC, BCG be equal, and the four pairs of corresponding angles be equal, and the two interior angles on the same side be supplementary.

Because

EBC=BCH,

(Hyp.) and EBC= ABF being vertically opposite angles, (Th. 4.)

therefore also their supplements, the angles ABE and BCG are equal.

Therefore also the angles which are respectively vertically opposite to these angles are equal,

Again, because the angle EBC= the alternate angle BCH add to each the angle CBF;

therefore the two angles EBC, CBF are equal to the two CBF, BCH;

but the two EBC, CBF are together equal to two right angles;

therefore the two CBF, BCH are together equal to two right angles.

And in the same way it may be shewn that if two corresponding angles are given equal, or if two interior angles on the same side are supplementary, then the alternate angles will be equal.

COR.

Hence if two parallel straight lines are intersected by a third straight line, the corresponding angles are equal, and the interior angles on the same side are supplementary; and conversely.

THEOREM 24.

Straight lines which are parallel to the same straight line are parallel to one another*.

Part. En. Let A and B be each of them parallel to X,

it is required to prove that A is parallel to B.

Proof. If A intersected B, then two intersecting lines,

A

B

A, B would each be parallel to a third line X, which is impossible, by Axiom 5.

Therefore

that is,

A does not intersect B,

A is parallel to B.

* Euclid, 1. 30.

Q. E. D.