NOTE III. On Euclid's definition of an Angle.

Euclid directs us to regard an angle as the inclination of two straight lines to each other, which meet, but are not in the same straight line.

Thus he does not recognize the existence of a single angle equal in magnitude to two right angles.

Euclid's definition may be extended with advantage in the following terms:

DEF. Let WQE be a fixed straight line, and QP a line which revolves about the fixed point Q, and which at first coincides with QE.

Then, when QP has reached the position represented in the diagram, we say that it has described the angle EQP.

When QP has revolved so far as to coincide with QW, we say that it has described an angle equal to two right angles.

Hence we may obtain an easy proof of Prop. XIII.; for whatever the position of PQ may be, the angles which it makes with WE are together equal to two right angles.

Again, in Prop. xv. it is evident that AED

since each has the same supplementary

AEC.

= L BEC,

We shall shew hereafter how this definition may be extended, so as to embrace angles greater than two right angles.

PROPOSITION XVI. THEOREM.

If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Let the side BC of a ABC be produced to D.

Then must ▲ ACD be greater than either z CAB or ↳ ABC.
Bisect AC in E, and join BE.

Produce BE to F, making EF= BE, and join FC.

Then in AS BEA, FEC,

:: BE=FE, and EA = EC, and ▲ BEA= ▲ FEC, 1. 15.

.. LECF L EAB.

I. 4.

Now

that is,

and

Similarly, if AC be produced to G it may be shewn that
L BCG is greater than ABC:

LBCG LACD;

.. LACD is greater than ABC.

I. 15.

Q.E.D.

Ex. 1. From the same point there cannot be drawn more than two equal straight lines to meet a given straight line.

Ex. 2. If, from any point, a straight line be drawn to a given straight line making with it an acute and an obtuse angle, and if, from the same po'nt, a perpendicular be drawn to the given line; the perpendicular will fall on the side of the acute angle.

PROPOSITION XVII. THEOREM.

Any two angles of a triangle are together less than two right angles.

Then must any two of its Ls be together less than two

rt. Ls.

Then

Produce BC to D.

LACD is greater than ABC.

I. 16.

.. LS ACD, ACB are together greater than 48 ABC, ACB. LS ACD, ACB together two rt. 4 s.

But

I. 13.

.. LS ABC, ACB are together less than two rt. ≤ s. Similarly it may be shewn that 48 ABC, BAC and also that 8 BAC, ACB are together less than two rt. 4 s.

NOTE IV. On the Sixth Postulate.

Q. E. D.

We learn from Prop. 17 that if two straight lines BM and CN, which meet in A, are met by another straight line DE in the points O, P,

the angles MOP and NPO are together less than two right angles.

The sixth postulate asserts that if a line DE meeting two other lines BM, CN makes MOP, NPQ, the two interior

angles on the same side of it, together less than two right angles, BM and CN shall meet if produced on the same side of DE on which are the angles MOP and NPO.

Thus Postulate 6 is the converse of Proposition XVII.

We shall explain hereafter why the Postulate cannot be proved as readily as the Proposition.

PROPOSITION XVIII. THEOREM.

If one side of a triangle be greater than a second, the angle opposite the first must be greater than that opposite the second.

B

D

In ▲ ABC, let side AC be greater than AB.

Then must ABC be greater than ACB.

From AC cut off AD=AB, and join BD.

And CD, a side of ▲ BDC, is produced to A,

.. LADB is greater than ACB;

.. also ABD is greater than ▲ ACB.

I. A.

I. 16.

Much more is LABC greater than ACB.

Q. E. D.

Ex. Shew that if two angles of a triangle be equal the

sides which subtend them are equal also (Eucl. 1. 6).

PROPOSITION XIX. THEOREM.

If one angle of a triangle be greater than a second, the side opposite the first must be greater than that opposite the second.

B

In ▲ ABC, let ABC be greater than ▲ ACB.
Then must AC be greater than AB.

For if AC be not greater than AB,

AC must either = AB, or be less than AB.

▲ ABC would = ▲ ACB, which is not the case.

And AC cannot be less than AB, for then

I. A.

I. 18.

▲ ABC would be less than ACB, which is not the case;

.. AC is greater than AB.

Q. E. D.

Ex. 1. In an obtuse-angled triangle, the greatest side is opposite the obtuse angle.

Ex. 2. BC, the base of an isosceles triangle BAC, is produced to any point D; shew that AD is greater than AB.

Ex. 3. The perpendicular is the shortest straight line, which can be drawn from a given point to a given straight line; and of others, that which is nearer to the perpendicular is less than the more remote.