13. All right angles are equal to one another.

14. Magnitudes which coincide throughout their whole extent, are equal.

POSTULATES.

35. A POSTULATE is a self-evident problem; such as, 1. That a straight line may be drawn from one point to another.

2. That a straight line may be produced to any length. 3. That a straight line may be drawn through a given point parallel to another straight line.

4. That a perpendicular to a given straight line may be drawn from a point either within or without the line.

5. That an angle may be described equal to any given angle.

PROPOSITIONS.

36. A DEMONSTRATION is a course of reasoning by which a truth becomes evident.

37. A PROPOSITION is something proposed to be demonstrated, or to be performed.

A proposition is said to be the converse of another, when the conclusion of the first is used as the supposition in the second.

38. A THEOREM is something to be demonstrated.

39. A PROBLEM is something to be performed.

40. A LEMMA is a proposition preparatory to the demonstration or solution of a succeeding proposition.

41. A COROLLARY is an obvious consequence deduced from one or more propositions.

42. A SCHOLIUM is a remark made upon one or more preceding propositions.

43. An HYPOTHESIS is a supposition, made either in the

enunciation of a proposition, or in the course of a demonstration.

PROPOSITION I.-THEOREM.

44. The adjacent angles which one straight line makes by meeting another straight line, are together equal to two right angles.

Let the straight line D C meet AB, making the adjacent angles ACD, DCB; these angles together will be equal to two right angles.

A

E

D

C

B

From the point C suppose CE to be drawn perpendicular to AB; then the angles ACE and ECB will each be a right angle (Art. 15). But the angle A CD is composed of the right angle A CE and the angle ECD (Art. 34, Ax. 9), and the angles ECD and DCB compose the other right angle, ECB; hence the angles ACD, DCB together equal two right angles.

45. Cor. 1. If one of the angles ACD, DCB is a right angle, the other must also be a right angle.

46. Cor. 2. All the successive angles, BA C, CAD, DA E, EAF, formed on the same side of a straight line, BF, are equal, when taken together, to two right angles; for their sum is equal to that of the two adjacent angles, BAC, CAF.

C

D

E

B

F

A

PROPOSITION II.-THEOREM.

47. If one straight line meets two other straight lines at a common point, making adjacent angles, which together are equal to two right angles, the two lines form one and the same straight line.

Let the straight line DC meet the two straight lines A C, CB at the common point C, making the adjacent angles ACD, DCB together equal to two right angles; then the lines AC and CB will form one and the same straight line.

A

C

D

BE

If CB is not the straight line AC produced, let CE be that line produced; then the line ACE being straight, the sum of the angles ACD and DCE will be equal to two right angles (Prop. I.). But by hypothesis the angles ACD and DCB are together equal to two right angles; therefore the sum of the angles A CD and DCE must be equal to the sum of the angles ACD and DCB (Art. 34, Ax. 2). Take away the common angle A CD from each, and there will remain the angle D C B, equal to the angle DCE, a part to the whole, which is impossible; therefore CE is not the line A C produced. Hence AC and CB form one and the same straight line.

PROPOSITION III.-THEOREM.

48. Two straight lines, which have two points common, coincide with each other throughout their whole extent, and form one and the same straight line.

Let the two points which are common to two straight lines be A and B.

F

E

A

B C

The two lines must coincide between the points A and B, for otherwise there would be two straight lines between A and B, which is impossible (Art. 34, Ax. 11).

Suppose, however, that, on being produced, the lines begin to separate at the point C, the one taking the direc

tion CD, and the other CE. From the point C let the line CF be drawn, making, with CA, the right angle ACF. Now, since ACD is a straight line, the angle FCD will be a right angle (Prop. I. Cor. 1); and since ACE is a straight line, the angle FCE will also be a right angle; therefore the angle FCE is equal to the angle FCD (Art. 34, Ax. 13), a part to the whole, which is impossible; hence two straight lines which have two points common, A and B, cannot separate from each other when produced; hence they must form one and the same straight line.

C

49. When two straight lines intersect each other, the opposite or vertical angles which they form are equal. Let the two straight lines AB, CD intersect each other at the point E; then will the angle AEC be equal to the angle DEB, and the angle CEB to AED.

E

A

B

D

For the angles AEC, CEB, which the straight line CE forms by meeting the straight line AB, are together equal to two right angles (Prop.I.); and the angles CEB, BED, which the straight line BE forms by meeting the straight line CD, are equal to two right angles; hence the sum of the angles AEC, CEB is equal to the sum of the angles CEB, BED (Art. 34, Ax. 1). Take away from each of these sums the common angle CE B, and there will remain the angle A E C, equal to its opposite angle, BED (Art. 34, Ax. 3).

In the same manner it may be shown that the angle CEB is equal to its opposite angle, A ED.

50. Cor. 1. The four angles formed by two straight lines intersecting each other, are together equal to four. right angles.

51. Cor. 2. All the successive angles formed by any number of straight lines meeting at a common point, are together equal to four right angles.

PROPOSITION V.-THEOREM.

52. If two triangles have two sides and the included angle in the one equal to two sides and the included angle in the other, each to each, the two triangles will be equal.

In the two triangles ABC, DEF, let the side A B be equal to the side DE, the side A C to the side DF, and the angle A to the angle D ;

B

A

CE

then the triangles ABC, D E F will be equal.

D

F

Conceive the triangle ABC to be applied to the triangle DEF, so that the side A B shall fall upon its equal, D E, the point A upon D, and the point B upon E; then, since the angle A is equal to the angle D, the side AC will take the direction DF. But AC is equal to DF; therefore the point C will fall upon F, and the third side BC will coincide with the third side E F (Art. 34, Ax. 11). Hence the triangle A B C coincides with the triangle D EF, and they are therefore equal (Art. 34, Ax. 14).

53. Cor. When, in two triangles, these three parts are equal, namely, the side A B equal to D E, the side A C equal to D F, and the angle A equal to D, the other three corresponding parts are also equal, namely, the side BC equal to E F, the angle B equal to E, and the angle C equal to F.

PROPOSITION VI.-THEOREM.

54. If two triangles have two angles and the included side in the one equal to two angles and the included side in the other, each to each, the two triangles will be equal.