II. THEOREMS.

16. We shall now state some important geometrical facts which it will be advantageous to remember. These facts, with many others, are demonstrated in Euclid's Elements of Geometry, that is they are shewn by strict reasoning to be necessarily true: and all who have the opportunity should study Euclid's demonstrations. But the present work may be used by those who have not yet applied themselves to demonstrative Geometry; and so they may for some time be satisfied with understanding the meaning of the statements which we shall make and committing them to memory.

When the demonstration is of a very simple character we shall however give the substance of it, and an attempt to master this, even if at first not entirely successful, will prove very beneficial. Some of the statements for which we give no demonstration will appear almost self-evident; others may be verified by repeated practical measurement; so that at last a confidence in the truth of all may be gained, approaching to absolute conviction.

17. We have selected only a few of the most important geometrical facts out of the large collection which has been formed by the investigations of mathematicians; but these specimens will be sufficient to suggest some idea of the extent and variety of the results which follow by strict connexion from a few elementary principles, and may tempt the student to increase his knowledge of the subject hereafter.

It will be found that Arts. 18...21 relate to angles, Arts. 22...27 relate to triangles, Arts. 28...30 relate to the equivalence of areas, Arts. 31...33 relate to properties of the circle, and Arts. 34...38 to similar triangles.

18. Let the straight line AB make with the straight line CD on one side of it the angles ABC and ABD: these angles will be together equal to two right angles.

For let BE be at right angles to DC. Then the angle ABD is the sum of the angles ABE and EBD; so that the

sum of the two angles ABC and ABD is equal to the sum of the three angles ABC, ABE, and EBD.

But the angle EBD is a right angle; and the sum of the angles ABC and ABE is EBC, which is also a right angle.

Therefore the angles ABC and ABD are together equal to two right angles.

19. Let two straight lines AB and CD cut one another at E; the angle AEC will be equal to the angle BED, and the angle AED will be equal to the angle BEC.

For the angles AEC and CEB are together equal to two right angles, and the angles CEB and BED are also equal to two right angles, by Art. 18. Thus the angles AEC and CEB are together

A

equal to the angles CEB and BED. Therefore the angle AEC must be equal to the angle BED.

In a similar manner we can shew that the angle AED is equal to the angle BEC.

The angles AEC and BED are called vertically opposite angles; and so also are the angles AED and BEC.

20. Let the straight line EF cut the parallel straight lines AB, CD: the angle EGB will be equal to the angle GHD, and the two angles BGH and GHD will be together equal to two right angles.

A

F

B

21. Since the angle EGB is equal to the angle AGH by Art. 19, it follows from the first part of the preceding Article that the angle AGH is equal to the angle GHD: these angles are called alternate angles.

So also the angle BGH is equal to the alternate angle GHC.

22. Let BC a side of the triangle ABC be produced to D; the exterior angle ACD will be equal to the two interior and opposite angles.

For suppose CE to be parallel to BA. Then the angle ECD is equal to the angle ABC by Art. 20; and the angle ACE is equal to the angle BAC by Art. 21. Thus the whole

B

angle ACD is equal to the sum of the two angles ABC and BAC.

23. The three angles of any triangle are together equal to two right angles.

For by Art. 22 the sum equal to the angle ACD. Thus the sum of the three angles ABC, BAC, and ACB is equal to the sum of the two angles ACD and ACB; that is to two right angles, by Art. 18.

of the angles ABC and BAC is

24. If two sides of a triangle are equal the angles opposite to them will also be equal.

25. If two angles of a triangle are equal the sides opposite to them will also be equal.

26. If two sides of one triangle are equal to two sides of another, each to each, and the angle contained by the two sides of the one equal to the angle contained by the two sides of the other, the triangles will be equal in all respects.

27. If two angles of one triangle are equal to two angles of another, each to each, and the side adjacent to the two angles of the one equal to the side adjacent to the two angles of the other, the triangles will be equal in all respects.

28. A parallelogram is equivalent to a rectangle on the same base and between the same parallels.

Let ABCD be a parallelogram, and ABEF a rectangle, on the same base AB, and between the same paral

It is in fact easy to admit that the triangle BEC is equal to the triangle AFD; and hence it fol- Å lows that ABCD is equiva

lent to ABEF.

E

Instead of saying that the parallelogram and the rectangle are between the same parallels, we may say that they have the same height: see Art. 13.

29. A triangle is equivalent to half a rectangle having the same base and height.

Let ABC be a triangle, and E ABDE a rectangle, on the same base AB, and having the same height: the triangle is equivalent to half the rectangle.

Let CF be the perpendicular

from C on AB. It is easy to ad- A

F

D

B

mit that the triangle BFC is equal to the triangle CDB, and that the triangle AFC is equal to the triangle CEA; and hence it follows that ABC is equivalent to half ABDE.

Hence two triangles which have the same base and equal heights are equivalent.

30. In any right-angled triangle the square described on the hypotenuse is equal to the sum of the squares described on the sides.

The figure represents a right-angled triangle having squares described on its hypotenuse and its sides: the largest square is equal in size to the sum of the other two. This statement is one of the most important in Geometry; and we will shew how its truth may be illus-. trated.

2

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Let BCDEFG be a figure composed of two squares placed side by side: take GH F equal to BC, and draw the straight lines CH and FH. Cut the whole figure out in paper or cardboard, and then divide it into the three pieces marked 1, 2, and 3. Fit the pieces together in the manner indicated by the figure HCAF. It will be found that a single square is thus obtained, each side being equal to FH.

Hence we see that the F square described on FH is equal to the sum of the squares described on FG and GH.