Fig. 23.

DEF, coincide, the one with the other, and are equal in all respects.

39. Corollary. When, in two triangles, these three things are equal, namely, BC = EF, B = E, and CF, we may thence infer that the other three are also equal, namely, AB = DE, AC DF, and A = D.

=

THEOREM.

40. One side of a triangle is less than the sum of the other two. Demonstration. The straight line BC (fig. 23), for example, is the shortest way from B to C (3); BC therefore is less than BA+AC.

Fig. 24.

Fig. 25.

41. If from a point ○ (fig. 24), within a triangle ABC, there be drawn straight lines OB, OC, to the extremities of BC, one of its sides, the sum of these lines will be less than that of AB, AC, the two other sides.

Demonstration. Let BO be produced till it meet the side AC in D; the straight line OC is less than OD + DC; to each of these add BO, and BO+OC < BO+OD + DC ; that is

BO+OCBD + DC.

Again, BD < BA + AD; to each of these add DC, and we shall have BD+DC < BA+AC. But it has just been shown that BO+OC < BD+DC, much more then is

42. If two sides AB, AC (fig. 25), of a triangle ABC, are equal to two sides DE, DF, of another triangle DEF, each to each; if, at the same time, the angle BAC, contained by the former, is greater than the angle EDF, contained by the latter; the third side BC of the first triangle, will be greater than the third side EF of the second.

Demonstration. Make the angle CAG = D, take AG = DE, and join CG, then the triangle GAC is equal to the triangle EDF (36), and therefore CG EF. Now there may be three cases, according as the point G falls without the triangle ABC, on the side BC, or within the triangle.

Case 1. Because GC < GI + IC, and AB < AI + IB, therefore GC+AB <GI + AI + IC + IB, that is,

GC+AB<AG +BC.

From one of these take away AB, and from the other its equal
AG, and there remains GCBC; therefore EF < BC.

Case 11. If the point G (fig. 26) fall upon the side BC, then Fig. 26. it is evident that GC, or its equal EF, is less than BC.

Case III. If the point G (fig. 27) fall within the triangle Fig 27. BAC, then AG + GC <AB + BC (41), therefore, taking away the equal quantities, AG, AB, we shall have GC < BC, or EF

BC.

THEOREM.

43. Two triangles are equal, when the three sides of the one are equal to the three sides of the other, each to each.

Demonstration. Let the side AB = DE (fig. 23,) AC = DF, Fig. 23 BC EF; then the angles will be equal, namely, A = D, B= E, and CF.

For, if the angle A were greater than the angle D, as the sides AB, AC, are equal to the sides DE, DF, each to each, the side BC would be greater than EF (42); and if the angle A were less than the angle D, then the side BC would be less than EF; but BC is equal to EF, therefore the angle A can neither be greater nor less than the angle D, that is, it is equal to it. In the same manner it may be proved, that the angle B = E, and that the angle C = F.

44. Scholium. It may be remarked, that equal angles are opposite to equal sides; thus, the equal angles A and D are opposite to the equal sides BC and EF.

45. In an isosceles triangle the angles opposite to the equal sides are equal.

Demonstration. Let the side AB= AC (fig. 28), then will the angle C be equal to B.

Draw the straight line AD from the vertex A to the point D the middle of the base BC; the two triangles ABD, ADC, will have the three sides of the one, equal to the three sides of the other, each to each, namely, AD common to both, AB = AC, by hypothesis, and BD = DC, by construction; therefore (43) the angle B is equal to the angle C.

46. Corollary. An equilateral triangle is also equiangular, that is, it has its angles equal.

Fig. 28,

ELEMENTS OF GEOMETRY.

Definitions and Preliminary Remarks.

1. GEOMETRY is a science which has for its object the measure of extension.

Extension has three dimensions, length, breadth, and thick

ness.

2. A line is length without breadth.

The extremities of a line are called points. A point, therefore, has no extension.

3. A straight or right line is the shortest way from one point

to another.

4. Every line, which is neither a straight line nor composed of straight lines, is a curved line.

Thus AB (fig. 1) is a straight line, ACDB is a broken line, or Fig. 1. one composed of straight lines, and AEB is a curved line.

5. A surface is that which has length and breadth, without thickness.

6. A plane is a surface, in which any two points being taken, the straight line joining those points lies wholly in that surface. 7. Every surface, which is neither a plane nor composed of planes, is a curved surface.

8. A solid is that which unites the three dimensions of extension.

9. When two straight lines, AB, AC, (fig. 2), meet, the quan- Fig. 2. tity, whether greater or less, by which they depart from each other as to their position, is called an angle; the point of meeting or intersection A, is the vertex of the angle; the lines AB, AC, are its sides.

An angle is sometimes denoted simply by the letter at the vertex, as A; sometimes by three letters, as BAC, or CAB, the letter at the vertex always occupying the middle place.

Fig. 29.

Fig. 30.

47. Scholium. From the equality of the triangles ABD, ACD, it follows, that the angle BAD DAC, and that the angle BDA=ADC; therefore these two last are right angles. Hence a straight line drawn from the vertex of an isosceles triangle, to the middle of the base, is perpendicular to that base, and divides the vertical angle into two equal parts.

In a triangle that is not isosceles, any one of its sides may be taken indifferently for a base; and then its vertex is that of the opposite angle. In an isosceles triangle, the base is that side. which is not equal to one of the others.

48. Reciprocally, if two angles of a triangle are equal, the opposite sides are equal, and the triangle is isosceles.

Demonstration. Let the angle ABC=ACB (fig. 29), the side AC will be equal to the side AB.

For, if these sides are not equal, let AB be the greater. Take BD = AC, and join DC. The angle DBC is, by hypothesis, equal to ACB, and the two sides DB, BC, are equal to the two sides AC, CB, each to each; therefore the triangle DBC is equal to the triangle ACB (36); but a part cannot be equal to the whole; therefore the sides AB, AC, cannot be unequal; . that is, they are equal, and the triangle is isosceles.

THEOREM.

49. Of the two sides of a triangle, that is the greater, which is opposite to the greater angle; and conversely, of the two angles of a triangle, that is the greater, which is opposite to the greater side.

Demonstration. 1. Let the angle C> B (fig. 30), then will the side AB, opposite to the angle C, be greater than the side AC, opposite to the angle B.

Draw CD making the angle BCD=B. In the triangle BDC, BD is equal to DC (48); but AD + DC > AC, and

AD+DC=AD + DB = AB, therefore AB>AC.

2. Let the side AB > AC, then will the angle C, opposite to the side AB, be greater than the angle B, opposite to the side AC. For, if C were less than B, then according to what has just been demonstrated we should have AB < AC, which is contrary to the hypothesis; and if C were equal to B, then it would