the order A'B'C". But by Assumption IX, AC is congruent to AC". Hence by Assumption VII, C"=C'. Hence {AB'C'}. By combining this theorem with Assumption IX we have the following

Corollary. If (A, B) and (A, C) are congruent respectively to (A',B') and (A',C'), and if C is on the ray AB and C' on the ray A'B', then (if BC) (B, C) is congruent to (B,' C').

Theorem 25.* (A, B) is congruent to (A, B).

Proof. By Assumption X, (A, B) is congruent to (B, A) and (B, A) is congruent to (A, B). Hence by Assumption. VIII, (A, B) is congruent to (A, B).

Theorem 26. If (A, B) is congruent to (C, D) then (C, D) is congruent to (A, B).

Proof. By Assumption VII there exists on the ray AB a point B' such that (C, D) is congruent to (4, B'). Hence by Assumption VIII (A, B) is congruent to (A, B′). by Theorem 25 and Assumption VII B' = B.

Hence

Corollary. If (A, B) is congruent to (C, D) and also to (E, F) then (C, D) is congruent to (E, F).

Proof. Since (4, B) is congruent to (C, D), (C, D) is congruent to (A, B). Since also (A, B) is congruent to (E, F) it follows by Assumption VIII that (C, D) is congruent to (E, F).

The word congruent was taken without definition as a relation between point-pairs. We now proceed to extend its significance by means of a definition.

Definition. A set of points [X] is congruent to a set of points [Y] if (1) every point X corresponds to one point Y in such a way that whenever (X1, X2) corresponds to (Y1, Y2), (X1, X2) is congruent to (Y1, Y2) and (2) every point Y is the correspondent of one point X.

This definition corresponds precisely to the intuitive conception of superposition. If two plane figures are represented by drawings on sheets of paper it is perfectly clear that a test. for their congruence is to lay one on top of the other. The

*Cf. Euclid, Common Notion 4.

† Cf. Euclid, Common Notion 1.

superposition with which we have to do in geometry is, however, a kind of intellectual matching of two figures together. The attention is transferred from one to the other and we try to see whether corresponding pairs of points are congruent. It would be perfectly feasible to substitute the word superposable for congruent in the definition above.

Theorem 27.* Any figure is congruent to itself. If a figure is congruent to a second figure the second figure is congruent to the first. Two figures congruent to the same figure are congruent to each other.

Theorem 28. Any point is congruent to any other point, any line to any other line, any ray to any other ray, any straight angle to any other straight angle.

Proof. That any point, A, is congruent to any point, B, is obvious from the wording of the definition.

Let AB and LM be any two rays. Let each point Y of the ray LM correspond to that point X of the ray AB which is such that (A, X) is con

gruent respectively to (4, X1) and (A, X2) and hence by the corollary of Theorem 24 (X1, X2) is congruent to (Y1, Y2). Hence the ray LM is congruent to the ray AB.

By applying like reasoning to the rays AB' and LM', which are the prolongations beyond A and L of the segments BA and LM respectively, we have that the straight angle BAB' is congruent to the straight angle MLM'. Hence the two lines AB and LM are also congruent.

Theorem 29. If (A, B) is congruent to (C, D) then the segment AB is congruent to the segment CD and the interval AB is congruent to the interval CD.

* Cf. Euclid Common Notions 1 and 4. By the word, figure, we mean any set of points.

Proof. Let A correspond to C and B to D and any point X of the segment AB correspond to that point Y of the ray CD such that (A, X) is congruent to (C, Y). By Theorem 24, Y is on the segment CD and by the corollary of the same theorem, (D, Y) is congruent to (B, X). By the corollary of Theorem 24, if X1X2 correspond to Y1Y2 then (X1, X2) is congruent to (Y1, Y2).

VII. CONGRUENCE OF ANGLES

In order to deal with the congruence of angles and other figures in a plane, we must introduce an additional assumption. Assumption XI. If A, B, C are three non-collinear points

Bo

1 B'o

FIG. 34.

and D is a point in the order {BCD}, and if A'B'C' are three non-collinear points and D' is a point in the order {B'C'D'} such that the point pairs (A, B), (B,C), (C, A), (B, D) are respectively congruent to (A', B'), (B', C′), (C', A′), (B′, D′), then (A, D) is congruent to (A', D').

Theorem 30.* Two angles BAC and 4MON are congruent in such a way that A corresponds to O if there are two points P and Q on the rays OM and ON such that the point pairs (A, B), (A, C) and (B, C) are respectively congruent to (0, P), (0, Q) and (P, Q).

Proof. If the points P and Q exist as stated let A correspond to 0, B to P and C to Q. The ray AC is congruent to the ray OQ and the ray AB to the ray OP. Hence to prove the angles congruent we need to show that if X1 is any point of the ray AC and X2 any point of the ray AB and Y1 and Y2 are the corresponding points of the rays ON and OM respectively, then (X1, X2) is congruent to (Y1, Y2).

1

Let B' and P' be points on the prolongations of BA and PO beyond A and O respectively such that (A, B′) is congruent to

* Cf. Euclid, I, 8.

(O, P'). Since (A, C), (C, B), (B, A), (B, B′) are congruent (Q, P), (P, 0), (P, P'), it follows by

respectively to (O, Q), Assumption XI that congruent to (P', Q).

(B'C) is

Now if

B

A

B'

FIG. 35.

Y

X2 and Y2 are points of the rays AB and OP respectively such that (A, X2) is congruent to (0, Y2), it follows, since (A, C), (C, B′), (B′, A), (B′, X2) are respectively congruent to (0, Q), (Q, P'), (P′, 0), (P′, Y2), that (C, X2) is congruent to (Q, Y2). In similar fashion we can prove that if X and Y1 are points of the rays AC and OQ respectively such that (A, X1) is congruent to (O, Y1), then (X1, X2) is congruent to (Y1, Y2). Definition. If B' is on the prolongation of the segment BA beyond A, the angle B'AC is said to be a supplement of 4BAC. If C' is a point on the prolongation of the segment CA beyond A, the angles CAB and 4C'AB' are said to be

vertical.

Corollary 1. Supplements of congruent angles are congruent. Corollary 2.* Vertical angles are congruent.

Definition. In a triangle ABC, the sides AB and BC are said to include 4ABC. The side AC and angle ABC are said to be opposite each to the other. The sides AB and BC are said to be adjacent to each other and to 4ABC.

Theorem 31.† If the sides of one triangle are congruent respectively to the sides of another triangle, the triangles are congruent.

Proof. Let the two triangles be ABC and A'B'C' and let the segments AB, BC, CA be congruent respectively to the segments A'B', B'C', C'A'. This determines a correspondence between the two triangles in which by Theorem 30 the angles at A, B, and C correspond to congruent angles at A'B'C'. But since 4ABC is congruent to A'B'C' it follows by

* Cf. Euclid, I, 15.

† Cf. Euclid, I, 8.

definition that if X and Y are any two points of the segments BA and BC respectively, and X and Y the corresponding points of the segments B'A' and B'C' respectively, then (X, I' ́) is congruent to (X, Y). Applying the same argument to the angles 4ACB A'C'B' and the

X

Χ

and
angles BAC and 4 B'A'C'
we have that the two tri-
angles are congruent.

The following theorems are proved similarly and are left as an exercise for the reader.

Theorem 32.* If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle the two triangles are congruent.

Theorem 33. If two sides of a triangle are congruent, the angles opposite them are congruent.

VIII. INTERSECTIONS OF CIRCLES

B

Definition. If O and Xo are two points of a plane a, then the set of points [X] of a such that (0, X) is congruent to (0, Xo) is called a circle. O is called its centre and any one of the intervals OX is called a radius. The two radii on any line. through O constitute a diameter. The points, except the points [X], on radii of the circle are said to be interior to the circle. The points of a not on radii are said to be exterior to the circle.

FIG. 37.

It can be proved that the interior and exterior points constitute two regions into which the plane a is decomposed by

* Cf. Euclid, I, 4.

† Cf. Euclid, I, 5.