The Teaching of Geometry Part 15

THEOREM. _Two lines in the same plane perpendicular to the same line cannot meet, however far they are produced._

This proposition is not in Euclid, and it is introduced for educational rather than for mathematical reasons. Euclid introduced the subject by the proposition that, if alternate angles are equal, the lines are parallel. It is, however, simpler to begin with this proposition, and there is some advantage in stating it in such a way as to prove that parallels exist before they are defined. The proposition is properly followed by the definition of parallels and by the postulate that has been discussed on page 127.

A good application of this proposition is the one concerning a method of drawing parallel lines by the use of a carpenter's square. Here two lines are drawn perpendicular to the edge of a board or a ruler, and these are parallel.

THEOREM. _If a line is perpendicular to one of two parallel lines, it is perpendicular to the other also._

This, like the preceding proposition, is a special case under a later theorem. It simplifies the treatment of parallels, however, and the beginner finds it easier to approach the difficulties gradually, through these two cases of perpendiculars. It should be noticed that this is an example of a partial converse, as explained on page 175. The preceding proposition may be stated thus: If _a_ is [perp] to _x_ and _b_ is [perp] to _x_, then _a_ is || to _b_. This proposition may be stated thus: If _a_ is [perp] to _x_ and _a_ is || to _b_, then _b_ is [perp]

to _x_. This is, therefore, a partial converse.

These two propositions having been proved, the usual definitions of the angles made by a transversal of two parallels may be given. It is unfortunate that we have no name for each of the two groups of four equal angles, and the name of "transverse angles" has been suggested.

This would simplify the statements of certain other propositions; thus: "If two parallel lines are cut by a transversal, the transverse angles are equal," and this includes two propositions as usually given. There is not as yet, however, any general sanction for the term.

THEOREM. _If two parallel lines are cut by a transversal, the alternate-interior angles are equal._

Euclid gave this as half of his Proposition 29. Indeed, he gives only four theorems on parallels, as against five propositions and several corollaries in most of our American textbooks. The reason for increasing the number is that each proposition may be less involved. Thus, instead of having one proposition for both exterior and interior angles, modern authors usually have one for the exterior and one for the interior, so as to make the difficult subject of parallels easier for beginners.

THEOREM. _When two straight lines in the same plane are cut by a transversal, if the alternate-interior angles are equal, the two straight lines are parallel._

This is the converse of the preceding theorem, and is half of Euclid I, 28, his theorem being divided for the reason above stated. There are several typical pairs of equal or supplemental angles that would lead to parallel lines, of which Euclid uses only part, leaving the other cases to be inferred. This accounts for the number of corollaries in this connection in later textbooks.

Surveyors make use of this proposition when they wish, without using a transit instrument, to run one line parallel to another.

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For example, suppose two boys are laying out a tennis court and they wish to run a line through _P_ parallel to _AB_. Take a 60-foot tape and swing it around _P_ until the other end rests on _AB_, as at _M_. Put a stake at _O_, 30 feet from _P_ and _M_. Then take any convenient point _N_ on _AB_, and measure _ON_. Suppose it equals 20 feet. Then sight from _N_ through _O_, and put a stake at _Q_ just 20 feet from _O_. Then _P_ and _Q_ determine the parallel, according to the proposition just mentioned.

THEOREM. _If two parallel lines are cut by a transversal, the exterior-interior angles are equal._

This is also a part of Euclid I, 29. It is usually followed by several corollaries, covering the minor and obvious cases omitted by the older writers. While it would be possible to dispense with these corollaries, they are helpful for definite reference in later propositions.

THEOREM. _The sum of the three angles of a triangle is equal to two right angles._

Euclid stated this as follows: "In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles." This states more than is necessary for the basal fact of the proposition, which is the constancy of the sum of the angles.

The theorem is one of the three most important propositions in plane geometry, the other two being the so-called Pythagorean Theorem, and a proposition relating to the proportionality of the sides of two triangles. These three form the foundation of trigonometry and of the mensuration of plane figures.

The history of the proposition is extensive. Eutocius (_ca._ 510 A.D.), in his commentary on Apollonius, says that Geminus (first century B.C.) testified that "the ancients investigated the theorem of the two right angles in each individual species of triangle, first in the equilateral, again in the isosceles, and afterwards in the scalene triangle." This, indeed, was the ancient plan, to proceed from the particular to the general. It is the natural order, it is the world's order, and it is well to follow it in all cases of difficulty in the cla.s.sroom.

Proclus (410-485 A.D.) tells us that Eudemus, who lived just before Euclid (or probably about 325 B.C.), affirmed that the theorem was due to the Pythagoreans, although this does not necessarily mean to the actual pupils of Pythagoras. The proof as he gives it consists in showing that _a_ = _a_', _b_ = _b_', and _a_' + _c_ + _b_' = two right angles. Since the proposition about the exterior angle of a triangle is attributed to Philippus of Mende (_ca._ 380 B.C.), the figure given by Eudemus is probably the one used by the Pythagoreans.

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There is also some reason for believing that Thales (_ca._ 600 B.C.) knew the theorem, for Diogenes Laertius (_ca._ 200 A.D.) quotes Pamphilius (first century A.D.) as saying that "he, having learned geometry from the Egyptians, was the first to inscribe a right triangle in a circle, and sacrificed an ox." The proof of this proposition requires the knowledge that the sum of the angles, at least in a right triangle, is two right angles. The proposition is frequently referred to by Aristotle.

There have been numerous attempts to prove the proposition without the use of parallel lines. Of these a German one, first given by Thibaut in the early part of the eighteenth century, is among the most interesting.

This, in simplified form, is as follows:

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Suppose an indefinite line _XY_ to lie on _AB_. Let it swing about _A_, counterclockwise, through [L]_A_, so as to lie on _AC_, as _X'Y'_. Then let it swing about _C_, through [L]_C_, so as to lie on _CB_, as _X''Y''_. Then let it swing about _B_, through [L]_B_, so as to lie on _BA_, as _X'''Y'''_. It now lies on _AB_, but it is turned over, _X'''_ being where _Y_ was, and _Y'''_ where _X_ was. In turning through [Ls]_A_, _B_, and _C_ it has therefore turned through two right angles.

One trouble with the proof is that the rotation has not been about the same point, so that it has never been looked upon as other than an interesting ill.u.s.tration.

Proclus tried to prove the theorem by saying that, if we have two perpendiculars to the same line, and suppose them to revolve about their feet so as to make a triangle, then the amount taken from the right angles is added to the vertical angle of the triangle, and therefore the sum of the angles continues to be two right angles. But, of course, to prove his statement requires a perpendicular to be drawn from the vertex to the base, and the theorem of parallels to be applied.

Pupils will find it interesting to cut off the corners of a paper triangle and fit the angles together so as to make a straight angle.

This theorem furnishes an opportunity for many interesting exercises, and in particular for determining the third angle when two angles of a triangle are given, or the second acute angle of a right triangle when one acute angle is given.

Of the simple outdoor applications of the proposition, one of the best is ill.u.s.trated in this figure.

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To ascertain the height of a tree or of the school building, fold a piece of paper so as to make an angle of 45. Then walk back from the tree until the top is seen at an angle of 45 with the ground (being therefore careful to have the base of the triangle level). Then the height _AC_ will equal the base _AB_, since _ABC_ is isosceles. A paper protractor may be used for the same purpose.

Distances can easily be measured by constructing a large equilateral triangle of heavy pasteboard, and standing pins at the vertices for the purpose of sighting.

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To measure _PC_, stand at some convenient point _A_ and sight along _APC_ and also along _AB_. Then walk along _AB_ until a point _B_ is reached from which _BC_ makes with _BA_ an angle of the triangle (60). Then _AC_ = _AB_, and since _AP_ can be measured, we can find _PC_.

Another simple method of measuring a distance _AC_ across a stream is shown in this figure.

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Measure the angle _CAX_, either in degrees, with a protractor, or by sighting along a piece of paper and marking down the angle. Then go along _XA_ produced until a point _B_ is reached from which _BC_ makes with _A_ an angle equal to half of angle _CAX_. Then it is easily shown that _AB_ = _AC_.

A navigator uses the same principle when he "doubles the angle on the bow" to find his distance from a lighthouse or other object.

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If he is sailing on the course _ABC_ and notes a lighthouse _L_ when he is at _A_, and takes the angle _A_, and if he notices when the angle that the lighthouse makes with his course is just twice the angle noted at _A_, then _BL_ = _AB_. He has _AB_ from his log (an instrument that tells how far a ship goes in a given time), so he knows _BL_. He has "doubled the angle on the bow" to get this distance.

It would have been possible for Thales, if he knew this proposition, to have measured the distance of the ship at sea by some such device as this:

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Make a large isosceles triangle out of wood, and, standing at _T_, sight to the ship and along the sh.o.r.e on a line _TA_, using the vertical angle of the triangle. Then go along _TA_ until a point _P_ is reached, from which _T_ and _S_ can be seen along the sides of a base angle of the triangle. Then _TP_ = _TS_. By measuring _TB_, _BS_ can then be found.

THEOREM. _The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side_.

If the postulate is a.s.sumed that a straight line is the shortest path between two points, then the first part of this theorem requires no further proof, and the second part follows at once from the axiom of inequalities. This seems the better plan for beginners, and the proposition may be considered as semiobvious. Euclid proved the first part, not having a.s.sumed the postulate. Proclus tells us that the Epicureans (the followers of Epicurus, the Greek philosopher, 342-270 B.C.) used to ridicule this theorem, saying that even an a.s.s knew it, for if he wished to get food, he walked in a straight line and not along two sides of a triangle. Proclus replied that it was one thing to know the truth and another thing to prove it, meaning that the value of geometry lay in the proof rather than in the mere facts, a thing that all who seek to reform the teaching of geometry would do well to keep in mind. The theorem might simply appear as a corollary under the postulate if it were of any importance to reduce the number of propositions one more.

If the proposition is postponed until after those concerning the inequalities of angles and sides of a triangle, there are several good proofs.

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