NOTE ON A THEOREM AND ITS CONVERSE.

The enunciation of a theorem can generally be divided into two parts (i) the data, or hypothesis, (ii) the conclusion.

For instance in Th. 1 the hypothesis is that BOC is a straight line; the conclusion is that the adjacent angles are supplementary. If the hypothesis and conclusion are interchanged a second theorem is obtained, which is called the converse of the first theorem.

Thus the converse of Th. 1 asserts that, if the adjacent angles are supplementary, BOD is a straight line. This is Th. 2.

It must not be assumed that the converses of all true theorems are true; e.g. "if two angles are vertically opposite, they are equal" is a true theorem, but the converse "if two angles are equal, they are vertically opposite" is not a true theorem.

TEx. 1. State the converses of the following: are they true?

(i) If two sides of a triangle are equal, then two angles of the triangle are equal.

(ii) If a triangle has one of its angles a right angle, two of its angles are acute.

(iii) London Bridge is a stone bridge.

(iv) A nigger is a man with woolly hair.

THEOREM 3. If two straight lines intersect, the vertically opposite angles are equal.

Data The two st. lines AOB, COD intersect at O.

Proof Since st. line OD stands on st. line AB,

.. LAOD + 4 DOB 2 rt. 4s,

=

and since st. line OB stands on st. line CD,

..L DOB + 4 BOC = 2 rt. s,

.. LAOD + DOB = L DOB + 2 BOC,

A

D

B

Fig. 3.

Th. 1.

Th. 1.

Q. E. D.

PARALLEL STRAIGHT LINES.

DEF. Parallel straight lines are straight lines in the same plane which do not meet however far they are produced in either direction.

Two lines taken at random in space will generally be neither parallel nor intersecting. They are said to be skew. Two skew lines are not in the same plane.

THEOREM 4. (i) When a straight line cuts two other straight lines, if a pair of alternate angles are equal, then the two straight lines are parallel.

(i) Data The st. line EF cuts the two st. lines AB, CD at E, F, forming the s a, b, c, d; and a = alternate ▲ d.

Proof [We shall prove that, if AB, CD meet when produced to one side, they must also meet when produced to the other side.]

Take up the part AEFC;

call it A'E'F'C'; and, turning it round in its own plane, apply it to the part DFEB so that E' falls on F and E'A' along FD.

Now if AB, CD are not parallel, they must meet when produced in one direction or the other. Suppose that EB and FD meet when produced towards B and D; then F'C' and E'A' also must meet when produced towards C' and A', i.e. FC and EA must meet when produced towards C and A.

.. if AB, CD meet when produced in one direction, they will also meet when produced in the other direction; but this is impossible, for two st. lines cannot enclose a space*.

.. AB, CD cannot meet however far they are produced in either direction.

.. AB and CD are parallel.

THEOREM 4. (ii, iii) When a straight line cuts two other straight lines, if

(ii) a pair of corresponding angles are equal,

or (iii) a pair of interior angles on the same side of the cutting line are together equal to two right angles,

then the two straight lines are parallel.

* One of Euclid's assumptions, or axioms.

(ii) Data The st. line GH cuts the two st, lines AB, CD forming the

COR. If each of two straight lines is perpendicular to a third straight line, the two straight lines are parallel to one another.

NOTE. If any one of the above facts (i), (ii), (iii) has been proved the other two can be deduced from that.

¶Ex. 2. Assuming Th. 4 (ii), prove Th. 4 (i) and (iii).

¶Ex. 3. Assuming Th. 4 (iii), prove Th. 4 (i) and (ii).

PLAYFAIR'S AXIOM. Through a given point one straight line, and one only, can be drawn parallel to a given straight line.

NOTE ON PLAYFAIR'S AXIOM.

Theorem 5, the converse of Th. 4, cannot be proved without the use of Playfair's or some equivalent axiom. This axiom is in accordance with everyday experience. It cannot however be proved by experiment to be true. It is conceivable that there might be more than one straight line through a given point parallel to a given straight line. Through a point O we can certainly draw more than one line which does not meet AB if produced to the edge of the paper; a whole bundle of such lines might be drawn, all comprised within a certain angle. If the paper is enlarged, some of these lines will meet AB if produced to the new limits of the paper: but there will still remain others which do not meet; these will be comprised within an angle smaller than before. If we take a very large sheet of paper (say as large as a field), the lines which fail to meet AB will all be comprised within a very small angle, and might be almost indistinguishable from one another. But we cannot experiment with an infinitely large piece of paper, and therefore we can never verify by drawing that there is only one parallel. Nor will any experiment prove this. For practical purposes however we are quite prepared to assume the axiom without proof.

But it is possible to proceed in a different way and to enquire what results follow if we assume that the axiom is not true. This problem was examined by a Russian named Lobatchewsky (17931856), who discovered that a sound system of geometry could be built up on this assumption. In his system Th. 5 is untrue (a case of the converse of a true proposition being untrue). Many other of our familiar Euclidean theorems naturally fall at the same time; among others the theorem that the angle-sum for a triangle is two right angles. It may be objected that the truth of this theorem can surely be established by measurement, and the matter in this way brought to a test. But we cannot measure any object with absolute accuracy; all that we can assert is that, in triangles of such a size that we can measure them, the difference between the angle-sum and 180° is too small to detect.

It is therefore admitted that this non-Euclidean geometry is self