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

DC : AC = AC : BC; that is, the other side AC of the right-angular triangle ADC, 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 in the solution of almost every geometrical problem; the beginner will therefore do well to render himself perfectly familiar with them.

RECAPITULATION OF THE TRUTHS CONTAINED IN

THE SECOND SECTION.

PART I.

Quest. 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 also equal to one another, the two triangles must be equal in all their parts, that is, they must 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; and the reverse is also true, namely : the greater angle is opposite to the greater side, and the greatest angle to the greatest side.

7. In a right-angular triangle the greatest side is opposite to the right angle.

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

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

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 of the triangle, 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 smaller angle of the triangle.

12. If from a point without a straight line a perpendicular is dropped 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 dropped from that point to 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 furthest from the perpendicular.

16. If a perpendicular is drawn to a straight line, 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 point in that perpendicular, are equal to one another.

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 such, that a straight line drawn from it to a given point in that perpendicular, is of a given length.

18. 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 given magnitude.

19. 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.

20. If the hypothenuse and one side of a right-angular triangle are equal to the hypothenuse and one of the sides of another right-angular triangle, each to each, the two right-angular triangles are equal.

21. 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 the two sides 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.

22. If in a parallelogram a diagonal is drawn, it divides the parallelogram into two equal triangles.

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

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

25. By one angle of a parallelogram the four angles are determined.

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

27. A quadrilateral, in which two sides are equal and parallel, is a parallelogram.

28. If from one of the vertices of a rectilinear figure, diagonals are drawn to all the other vertices, the figure is divided into as many triangles as it has sides less two.

29. The sum of all the angles in a rectilinear figure, is equal to as many times two right angles as the figure has sides less two.

*

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