extremity (or the point) B will fall upon D, and the two lines will coincide throughout their whole extent.

Again, we will apply the line EF to the line CH. Taking the line EF (think of it as a little rod which you can pick up and handle), put the point E upon G, and making the line EF take the same direction as CH, put the former upon the latter. Now, since EF is shorter than CH, the point (extremity) F will fall somewhere on the line CH, as at I. Therefore the lines do not coincide throughout their whole extent, and are not equal.

84. Prob.-To apply one plane angle to another.

SOLUTION.-First we will apply one angle to another equal angle. Thus, to

E

B

N

H

M

FIG. 63.

apply BAC to the equal angle EDF. Take the angle BAC (think of it as if it were two little rods put firmly together at this angle, and so that you could pick them up and handle them), and placing the vertex (point) A upon the vertex (point) D, make the side AC take the direction DF. As AC happens to be longer than DF, the extremity C will fall beyond F, at some point, as O. But we do not care for this, as the size of an angle does not depend upon the length of the sides. Now, while A lies on D, and the line AC on DF, let the line AB be conceived as lying in the plane of the paper also (i. e., on it). Since the angle BAC is equal to EDF, the line AB will take the direction DE, and will fall on it, though the point B will fall somewhere beyond E, as at N, as AB chances to be longer than DE. The two angles therefore coincide, and are equal. [Notice carefully just what is meant by saying that the angles are equal. We do not mean that the sides are of the same length, but that the opening between them is the same, i. e., that one is just as sharp a corner as the other.]

Queries.-If BAC were greater than EDF, and we should begin by putting A upon D, and make AC fall upon DF, where would AB fall, without the angle EDF or within it? If BAC were less than EDF, and we proceed as before, placing the vertex A on D, and AC on DF, would AB fall without EDF or within it?

Again, let us attempt to apply the angle HCI to LKM. Placing the vertex G on the vertex K, making the side GI take the direction KM, and then bringing CH into the plane of the paper, the side GH will fall within the angle LKM (as in the line KR), since the angle HCI is less than LKM. The angles, therefore, do not coincide.

85. Prob.- When two triangles have two sides and the included angle of one equal to two sides and the included angle of the other, to apply one triangle to the other.

B

SOLUTION. In the two triangles ABC and DEF, let the angle A be equal to the angle D, the side AB = the side DE, and AC : = DF. We will apply the triangle ABC* to DEF. Take the triangle ABC and place the vertex A upon the vertex D, making the side AC take the direction DF. Since AC = DF, the extremity C will then fall on F. Now bring the triangle ABC A into the plane of DEF, keeping AC in DF, and the line AB will take the direction DE, since the angle A = the angle D. Again, as AB = DE, the extremity B will fall upon E. Thus we have placed ABC upon DEF, so that A falls upon D, Cupon F, and B upon E, and find that they exactly coincide.

FIG. 64.

last figure to Where will c

Ex. 1. Suppose you attempt to apply ABC in the DEF by placing B on D, and letting BC fall upon DF. fall? Measure it and find out. Which side will then fall nearly or quite on DE? Will it fall exactly on it? On which side will it fall? Can you make the triangles coincide (fit) in this way?

Ex. 2. Can you make the triangles in the last figure coincide by placing C upon D, and letting CA fall upon DF? Where will A fall? What line will fall on or near DE? Will it fall without DE, or within ?

=

с

E

Ex. 3. Construct two isosceles triangles, as ACB and DEF, in which ACCB DE EF. Can you apply DEF to ABC by putting D upon A? Describe the process. Can you put D upon A and DE upon AB, and make the triangles coincide? Can you make the A triangles coincide by putting F upon A?

BD

FIG. 65.

If so, describe the process. Can you make them coincide by putting E upon A? If not, point out the difficulties.

Think of ABC as made of little rods, so that you can pick it up and place it upon DEF in the manner described.

+ It will make it clearer if the pupil thinks of ABC, at this stage of the operation, as having the side AC on DF, but the angle B not down on the paper; just as if he were to cut out ABC, and set the edge AC on the line DF, and afterward bring the triangle ABC down on to DEF, keeping the edge AC on the line DF.

The teacher must insist upon the figures being drawn, and that accurately, according to rule.

A

FE

Ex. 4. Construct two equal trapeziums,* as ABCD and EFCH, and describe the process of applying one to the other.

SOLUTION.-I will apply EFGH to ABCD. As the angle E is equal to the angle B, I will begin by putting the vertex E on B, and making EH fall upon G BC. Since EH BC, H will fall on C. Now, as angle Hangle C, HG will take the direction (fall on) CD; and since HG CD, G will fall on D. Again, as CD, GF will take the direction DA; and since GFDA, F will fall on A. Finally, as F = A, FE will take the direction AB; and since

=

AB, E will fall on B, as it ought, since I started by conceiving E as placed on B.

Ex. 5. Describe the application of ABCD in the last figure to EFGH, by beginning with C upon H.

Ex. 6. Having two equal equilateral triangles, can you apply one to the other by beginning indifferently with any one angle of one upon any one angle of the other? Draw two such triangles, and go through with the details of the application.

86. Prob.-Given two triangles with two angles and the included side of the one respectively equal to two angles and the included side of the other, to apply one triangle to the other.

E

SOLUTION.-[The pupil should first draw any triangle, as ABC. Then make a line DF equal to AB, and at the extremities D and F make angles, as D and F, respectively equal to A and B. This is preliminary.] Having the two triangles ABC and DEF, in which A = D, B = F, and AB = DF, B I propose to apply one to the other. I will apply ABC to DEF. Taking ABC, I place A upon D, and make AB take the direction and fall upon DF. Since AB = DF, B will fall upon F. Now keeping the line AB in DF, I conceive the triangle ABC to come into the plane of DEF. Since A = D, the side AC will take the direction DE, and the extremity C of AC will fall somewhere in the line DE, or in DE produced. Also, since BF, the line BC will take the direction FE, and the extremity C of BC will fall somewhere in FE or FE produced. Finally, as C falls in DE and

FIG. 67.

* The teacher must insist upon the figures being drawn, and that accurately, according to Fule.

FĖ both, it must be at E, their intersection. Thus I find that the triangle ABC, when applied to DEF, coincides with it throughout..

Ex. 1. Given the two triangles DEF and ABC, in which DE=AB, D= A, but E>B; show how an attempt to apply one to the other fails.

=

SOLUTION. Since angle D = angle A,* I apply the vertex D to the vertex A, and make DE take the direction AB. As DE AB, E will fall on B, and the sides DE and AB will coincide. Again, since DA, the side DF will take the direction AC when the planes of the triangles coincide; and the extremity F will fall in AC, or in AC produced (really in AC produced, in this case). Finally, since E> B, EF will fall to the right of BC, and the application fails.

E

B

FIG. 68.

D

Ex. 2. Construct two trapeziums with their respective sides equal, as AC HE, AB HC, BDCF, and CD = EF, but with their angles unequal; and show how an attempt to apply one to the other fails.

Ex. 3. If the sides of two trapeziums, as in the last figure, are equal, and two of the angles including a side in one are respectively equal to the corresponding angles in the other, as A = H, and B = G, can one be applied to the other? If so, give the details of the process.

C

A

B

E

H

G

FIG. 69.

SECTION VI

ABOUT SIMILAR FIGURES, ESPECIALLY TRIANGLES.

87. Similar Figures are such as are shaped alike—i. e., have the same form.

A more scientific definition is, Similar Figures are such as have their angles respectively equal, and their homologous (correspond ing) sides proportional.

* Be careful to distinguish between the vertex, which is a point, and the angle, which is the opening between the lines.

88. Homologous, or Corresponding Sides of similar figures, are those which are included between equal angles in the respective figures.

IN SIMILAR TRIANGLES, THE HOMOLOGOUS SIDES ARE THOSE OPPOSITE THE EQUAL ANGLES.

ILL.-The triangles ABC and DEF are similar, for they are of the same

shape. But it is easy to see that ABC is not similar to IHK or MON. The pupil should notice that AD, CF, and B = E. Also, side e is 13 times b, side ƒ is 1 times c, and side d is 1 times a; so that fceb, and f:cd: a, and dae : b. Now there are no such relations existing between the parts of ABC and IHK. The angles B and K are nearly equal, but A is much larger than H, and C is

Again, as to their sides, IH is a
These two triangles are, there-

smaller than 1. So these triangles are not mutually equiangular, i. e., each angle in one has not an equal angle in the other. little less than AC, but HK is greater than AB. fore, not similar.

In the similar triangles ABC and DEF, b is homologous with e, since they are opposite the equal angles B and E. For a like reason a is homologous with d, and c with f. It may also be observed, that the shortest sides in two similar triangles are homologous with each other; the longest sides are also homologous with each other, and the sides intermediate in length are homologous with each other.

Ex. 1. Can a scalene triangle be similar to an isosceles triangle? Can an obtuse angled triangle be similar to a right angled triangle? Ex. 2. Are all squares similar figures?

SUG. First, are the angles equal? Second, is any one side of one square to some side of another square as a second side of the first is to a second side of the second, etc.?

Ex. 3. A farmer has two fields, each of which has 4 sides and 4 right angles. The first field is 20 rods by 50, and the other 40 by 80. Are they similar?

SUG. Are they mutually equiangular? Then are the lengths in the same ratio as the widths? If they are not similar, how long would the second have to be in order to make them similar? Draw two such figures, and see if they look alike in shape