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meets the side AQ and does not meet the side AP in triangle APQ. Therefore (by II 4) it meets the side PQ.

32. Convention. Using the notation of 31, we say the points A, A' lie on one and the same side of the plane a, and the points A, B lie on different sides of the plane a.

Ex. 1. A straight cannot traverse more than 4 of the 7 regions of the plane determined by the straights of the sides of a triangle.

Ex. 2. Four coplanar straights crossing two and two determine 6 points. Choosing 4 as vertices we can get two convex quadrilaterals, one of which has its sides on the straights.

Ex. 3. Each vertex of an n-gon determines with the others (1) straights. So together they determine n(n-1)/2.

Ex. 4. How many diagonals in a polygon of ʼn sides. Ex. 5. What polygon has as many diagonals as sides?

CHAPTER III.

CONGRUENCE.

III. The third group of assumptions: assumptions of

congruence.

33. The assumptions of this group make precise the idea of congruence.

34. Convention. Sects stand in certain relations to one another, for whose description the word congruent especially serves us.

III 1. If A, B are two points on a straight a, and A' a point on the same or another straight a', then we can find on the straight a' on a given ray from A' always one and only one point B' such that the sect AB is congruent to the sect A'B'.

We write this in symbols AB=A'B'.

Every sect is congruent to itself, i.e., always AB =AB. The sect AB is always congruent to the sect BA, i.e., AB=BA.

We also say more briefly, that every sect can be taken on a given side of a given point on a given straight in one and only one way.

III 2. If a sect AB is congruent as well to the sect A'B' as also to the sect A"B", then is also A'B' con

gruent to the sect A"B", i.e., if AB=A'B' and AB=A"B", then is also A'B' A"B".

III 3. On the straight a let AB and BC be two sects without common points, and furthermore A'B' and B'C' two sects on the same or another straight, likewise without common points; if then AB=A'B' and BC= B'C', so always also AC=A'C'.

35. Definition. Let a be any plane and h, k any two distinct rays in a going out from a point O, and pertaining to different straights. These two rays

AK

W

h

FIG. II.

h, k we call an angle, and des

ignate it by (h, k) or ☀ (k, h). The rays h and k, together with the point O, separate the other points of the plane a into two regions of the following character: if A is a point of the one region and B of the other region, then every sect-train which joins A with

B, goes either through O or has with h or k at least one point in common; on the contrary if A, A' are points of the same region, then there is always a sect-train which joins A with A' and neither goes through O nor through a point of the rays h, k.

One of these two regions is distinguished from the other because each sect which joins any two points of this distinguished region always lies wholly

in it; this distinguished region is called the interior of the angle (h, k) in contradistinction from the other region, which is called the exterior of the angle (h, k). The interior of (h, k) is wholly on the same side of the straight h as is the ray k, and altogether on the same side of the straight k as is the ray h.

The rays h, k are called sides of the angle, and the point O is called the vertex of the angle.

III 4. Given any angle (h, k) in a plane a and a straight a' in a plane a', also a determined side of a' on a'. Designate by h' a ray of the straight a' starting from the point O'; then there is in the plane a' ONE AND ONLY ONE ray k' such that the angle (h, k) is congruent to the angle (h', k'), and likewise all interior points of the angle (h', k') lie on the given side of a'.

Every angle is congruent to itself, i.e., always (h, k) = 4 (h, k).

The angle (h, k) is always congruent to the angle (k, h), i.e., ☀ (h, k) = ¥ (k, h).

We say also briefly, that in a given plane every angle can be set off towards a given side against a

I

given ray, but in a uniquely determined way. There is one and only one such angle congruent to a given angle. We say an angle so taken is uniquely determined.

III 5. If an angle (h, k) is congruent as well to the angle (h', k') as also to the angle (h", k''), then is also the angle (h', k') congruent to the angle (h", k'); i.e., if (h, k) = 4 (h′, k') and 4 (h, k) = ¥ (h'', k''), then always (h', k'′) = ¥ (h′′, k').

36. Convention. Let ABC be any assigned triangle; we designate the two rays going out from A through B and C respectively by h and k. Then the angle (h, k) is called the angle of the triangle ABC included by the sides AB and AC or opposite the side BC. It contains in its interior all the inner points of the triangle ABC and is designated by BAC or A.

III 6. If for two triangles ABC and A'B'C' we have the congruences

AB=A'B', AC=A'C', 4 BAC = B'A'C',

then always are fulfilled the congruences

ABC = A'B'C' and

ACB = A'C'B'.

Deductions from the assumptions of congruence.

37. Convention. Suppose the sect AB congruent to the sect A'B'. Since, by assumption III 1, also the sect AB is congruent to AB, so follows from III 2 that A'B' is congruent to AB; we say: the two sects AB and A'B' are congruent to one another. 38. Convention. Suppose (h, k) = 4 (h', k').