angle in the one triangle, are in proportion to the two sides which include that angle in the other triangle.

3dly. When the three sides of one triangle are in proportion to the three sides of another.

ABD and ACD, into which the whole triangle is divided, bear to each other, and to the whole triangle ABC itself?

A. The two triangles, ABD and ACD, are similar to each other, and to the whole triangle ABC.

Q. How can you prove this?

A. The triangle ABD is similar to the whole triangle ABC, because the two triangles being both right-angled, and having the angle at x common, have two angles in one triangle, respectively, equal to two angles in the other (page 73, case 1st); and for the same reason is the triangle ACD similar to the whole triangle ABC (both being right-angled, and having the angle y common); and as each of the two triangles, ABD, ACD, is similar to the whole triangle ABC, these two triangles must be similar to each other. (Truth II.)

Q. What important inferences can you draw from the principle you have just established?

A. 1st. In the two similar triangles, ABD and ACD, the sides which are opposite to the equal angles, must be

=

in proportion (condition 4th of geometrical similarity, page 70); and we shall therefore have the proportion BD: ADAD: DC ;*

that is, the perpendicular AD is a mean proportional between the two parts into which it divides the hypothenuse. (Theory of proportion, page 66.)

2dly. From the two similar triangles, ABC, ABD, we shall have the proportion

BD: ABAB : BC;

that is, the side AB, of the right-angled triangle ABC, is a mean proportional between the whole hypothenuse BC, and the part BD, cut off from it by the perpendicular AD.†

3dly. The two similar triangles, ACD and ABC, give the proportion

that is, the other side, AC, of the right-angled triangle ACD, is also a mean proportional between the whole hypothenuse and the other part, DC, cut off from it by the perpendicular AD.

Remark. The five last queries comprise one of the most important parts of Geometry. The principles contained in them are applied to the solution of almost every geometrical problem. The beginner will therefore do well to render himself perfectly familiar with them.

* The first ratio is formed by the two sides, BD and AD, of the triangle, ADB, of which BD is opposite to the angle z, and AD to the angle x; and the second ratio is formed by the two corresponding sides, AD, DC, of the triangle ADC; because the sides AD, DC, are opposite to the angles y and u, which are respectively equal to z and x.

†The part BD of the hypothenuse, situated between the extremity B of the side AB, and the foot D of the perpendicular AD, is sometimes called the adjacent segment to AB. (Legendre's Geometry, translated by Professor Farrar.)

RECAPITULATION OF THE TRUTHS CONTAINED IN THE SECOND SECTION.

PART I.

Ques. Can you now repeat the different principles respecting the equality and similarity of triangles, which you have learned in this section?

Ans. 1. If, in two triangles, two sides of the one are equal to two sides of the other, each to each, and the angles which are included by them are also equal to one another, the two triangles are equal in all their parts, that is, they coincide with each other throughout.

2. In equal triangles, that is, in triangles which coincide with each other, the equal sides are opposite to the equal angles.

3. If one side and the two adjacent angles in one triangle are equal to one side and the two adjacent angles in another triangle, each to each, the two triangles are equal, and the angles opposite to the equal sides are also equal.*

4. The two angles at the basis of an isosceles triangle are equal to one another.

5. If the three sides of one triangle are equal to the three sides of another, each to each, the two triangles coincide with each other throughout; that is, their angles are also equal, each to each.

6. In every triangle, the greater side is opposite to the greater angle, and the greatest side to the greatest angle. 7. In a right-angled triangle, the greatest side is opposite to the right angle.

This principle, though already demonstrated in the first section, is repeated here, in order to complete what is said on the equality of triangles.

8. When a triangle contains two equal angles, it also has two equal sides, and the triangle is isosceles.

9. If the three angles in a triangle are equal to each other, the sides are also equal, and the triangle is equilateral.

10. Any one side of a triangle is smaller than the sum of the two other sides.

11. If from a point within a triangle, two lines are drawn to the two extremities of one of the sides, the angle made by those lines is always greater than the angle of the triangle which is opposite to that side; but the sum of the two lines, which make the interior angle, is smaller than the sum of the two sides which include the angle of the triangle.

12. If from a point without a straight line, a perpendicular is let fall upon that line, and, at the same time, other lines are drawn obliquely to different points in the same straight line, the perpendicular is shorter than any of the oblique lines, and is therefore the shortest line that can be drawn from that point to the straight line.

13. The distance of a point from a straight line is measured by the length of the perpendicular, let fall from that point upon the straight line.

14. Of several oblique lines drawn from a point without a straight line, to different points in that straight line, that one is the shortest, which is nearest the perpendicular, and that one is the greatest, which is farthest from the perpendicular.

15. If a perpendicular is drawn to a straight line, then two oblique lines drawn from two points in the straight line, on each side of the perpendicular, and at equal distances from it, to any one point in that perpendicular, are equal to one another.

16. If a perpendicular is drawn to a straight line, there

is but one point in the straight line, on each side of the perpendicular, such, that a straight line drawn from it to a given point in that perpendicular, is of a given length.

17. If a perpendicular is drawn to a straight line, there is but one point in the straight line, on each side of the perpendicular, from which a line drawn to a given point in that perpendicular, makes with the straight line an angle of a required magnitude.

18. If two sides and the angle which is opposite to the greater of them, in one triangle, are equal to two sides and the angle which is opposite to the greater of them in another triangle, each to each, the two triangles coincide with each other in all their parts; that is, they are equal to each other.

19. If the hypothenuse and one side of a right-angled triangle, are equal to the hypothenuse and one side of another right-angled triangle, each to each, the two rightangled triangles are equal.

20. If in two triangles two sides of the one are equal to two sides of the other, each to each, but the angle included by the two sides in one triangle, is greater than the angle included by them in the other, the side opposite to the greater angle in the one triangle, is greater than the side opposite to the smaller angle in the other triangle. 21. Every parallelogram is, by a diagonal, divided into two equal triangles.

22. The opposite sides of a parallelogram are equal to each other.

23. The opposite angles in a parallelogram are equal to each other.

24. By one angle of a parallelogram all four angles are -determined.

25. A quadrilateral, in which the opposite sides are respectively equal, is a parallelogram.