Hence BC and EF coincide, therefore they are equal; (Ax. 8) and the triangles ABC, DEF will then coincide; therefore they are equal.

Also the angles ABC, DEF will coincide, and so must be equal; and the angles ACB, DFE, and therefore these are equal. Wherefore, if two triangles have two sides, &c.,

Which was to be demonstrated.

29. ON APPLICATIONS OF PROPOSITION IV.

This proposition is put to very frequent use; and it is required to become perfectly familiar with two sets of things belonging to its enunciation. These are, the number and nature of its conditions, and the number and nature of the consequences which are said to follow of necessity from those conditions. As we have explained already, these two sections of the enunciation are called its hypothesis and its thesis. Learners very often fail to become clear about either hypothesis or thesis; so we may specially notice that the hypothesis of Proposition IV. is threefold, and its thesis or conclusion is fourfold.

The hypothesis includes three equalities; namely,—

one side in one triangle equal to one side in the other,

a second side in one equal to a second side in the other,

and the contained angle of one triangle equal to the contained angle of the other.

Unless all these three conditions are granted, Proposition IV. proves nothing; and this fact is one which learners are rather apt to forget. The third is the condition most frequently overlooked; and, even when 'angle' is remembered, the word 'contained' is sometimes lost sight of.

The learner may easily make out for himself the four equalities contained in the conclusion.

Triangles which are equal in seven particulars, as ABC, DEF are shown to be in Prop. IV., are said to be equal in all respects. These seven particulars may be called 'elements' of the triangles; while the six angles and sides of a triangle are called its 'parts.'

30. TRIANGLES WITH PARTS IN COMMON.

As before explained, that which belongs to two things at the same time is said to be common to them.

In the figure to Proposition IV. the triangles are perfectly distinct one from the other; and they have therefore nothing in common. Now examine the figure given in Exercise VII. to No. 3, in which are two triangles ABC, ABD. Name the sides of each triangle; and you will observe that

one side belongs to both triangles. This side is therefore said to be common to both triangles.

In the course of future investigations, we shall meet with triangles having parts in common very frequently; in fact, more frequently than not. They will be found, for example, in all the eight propositions which come next, except one; and it will therefore be wise to take some preliminary practice in treating them.

The letters Q.E.D., which will be found at the end of Prop. V., are the initials of "quod erat demonstrandum"; and mean "which was to be demonstrated."

EXERCISE VII.

1. With respect to two triangles ABC, XYZ, let the hypothesis be that the sides AB, AC and angle BAC are equal to XY, XZ, and angle YXZ, each to each; write the fourfold conclusion.

2. Suppose in the triangles ABC, DEF the sides AB, DF are equal, also AC, DE, and angles BAC, EDF; state the conclusion so far as regards the other angles; that is, the third and fourth conclusions.

3. In the annexed figure, given that CA, DB are equal, name two sides of the triangle CAB which are respectively equal to two of the triangle DAB.

4. If the two angles CAB, DBA be also equal, state the four conclusions which may be drawn by the application of Prop. IV.

A

B

5. In the same figure, given CA, CB and the angle C equal to DB, DA and the angle D, each to each; deduce a threefold conclusion, and say why not fourfold.

6. In the annexed figure, given ABC is an isosceles triangle, and BD is made equal to CE; show AD, AE are equal.

7. Find three parts of the triangle ABE equal, each to each, to three of the triangle ACD.

D

8. Draw four inferences by aid of Prop. IV. 9. Suppose it granted in addition that the angles BDC and BEC are equal, prove by Prop. IV. that the angles at the base of the triangle ABC are equal.

B

10. In the annexed figure, state the parts com

E

mon to the two triangles BCA, DCA.

11. What is common to the triangles DAC and EAB?

12. Given, in these triangles, AD, AE equal, and also AB, AC equal; find three parts of the one respectively equal to three parts of the other.

D

B

13. Deduce a threefold conclusion, omitting that which relates to areas. 14. Given, in the same figure, AB, AC equal, and also AD, AE; prove BD, CE equal.

15. Given also BE, CD equal, and angles D, E; find three parts of the triangle DBC equal to three of the triangle ECB, each to each, without mentioning BC.

16. Deduce the conclusions relating to angles.

17. Take away the angle CBE from the angle ABE, and BCD from ACD; and state the remainders.

18. If ABE, ACD be equal, and also CBE, BCD, what will follow with regard to the two remainders ?

31. PROPOSITION V.-THEOREM.

The angles at the base of an isosceles triangle are equal to one another; and, if the equal sides be produced, the angles on the other side of the base shall be equal.

Let ABC be an isosceles triangle whose sides AB, ACare equal; and let AB, AC be produced to D and E.

Then the angles ABC, ACB shall be equal,

and also the angles DBC, ECB.

In BD take any point F;

(Post. 2)

from AE the greater cut off AG equal to AF the less; (I. 3) and join FC, GB.

B

F

A

D

Because AB is equal to AC (Hyp.), and AG to AF; (Constr.)

therefore, in the triangles ABG, ACF,

the two sides BA, AG are equal to the two CA, AF, each to each, and both pairs of sides contain the same angle FAG; therefore the bases BG and FC are equal,

(I. 4)

also the angles AGB, AFC respectively opposite to AB, AC, and the angles ABG, ACF opposite to AG, AF.

And, because the whole AF'is equal to the whole AG, (Constr.) of which the parts AB, AC are equal,

therefore the remainders BF and CG are equal.

Hence, in the triangles BFC, CGB,

(Ax. 3)

BF is equal to CG, and it was shown that FC is equal to GB, and also that the included angles BFC, CGB are equal:

therefore the angles BCF, CBG opposite to BF and CG are equal,

and also the angles CBF, BCG opposite to FC, GB.

And, since it has been demonstrated

(I. 4)

that the whole angles ABG and ACF are equal, and that parts of them, the angles CBG, BCF, are equal; therefore the remainders are equal, namely, the angles ABC, ACB,

which are the angles at the base of the triangle ABC. And CBF, BCG have been shown equal,

which are the angles on the other side of the base. Therefore, the angles at the base, &c.

(Ax. 3)

Q. E. D.

Corollary. Hence every equilateral triangle is also equiangular.

For the meaning of this term corollary,' see the remarks which follow. Note. To remember the order of proof in Prop. V., notice that it discusses a pair of large triangles and a pair of smaller ones; and applies Axiom 3 after each discussion.

32. ON COROLLARIES.

In the course of an extended piece of reasoning, such as that of Proposition V. or IV., we sometimes prove more truths than the one we have specially in view. More often still, other facts worthy of notice follow so easily from some part of the reasoning, or from the conclusion, that we perceive the truth of them at a glance, as soon as they are stated. For such inferences as these we have a name.

A Corollary to a proposition is an inference from its reasoning, or from its conclusion, which is so easy as to require little or no proof.

The abbreviation Cor.' stands for corollary.

As an example to the student, we will prove the corollary which occurs above to Prop. V., leaving as practice for himself such as will occur hereafter.

Theorem: Every equilateral triangle is also equiangular.

Let ABC be an equilateral triangle.

All its angles A, B, C shall be equal.
Because AB is equal to AC;

(Def. 24)

therefore the angle B is equal to the angle C. (I. 5)

Also, because BA is equal to BC;

therefore the angle A is equal to the angle C. And because CA is equal to AB;

therefore the angle A is equal to the angle B.

Wherefore the angles A, B, C are all equal.

33. CONVERSE THEOREMS.

B

Q. E. D.

When we compare two theorems in respect to their hypothesis and thesis, we sometimes find that the hypothesis of one is the thesis of the other; and vice versa. Such theorems are termed Converse Theorems. examples, compare the two theorems :

1. If two triangles be equilateral, they are also equiangular.

2. If two triangles be equiangular, they are also equilateral.

As

It will be desirable to say something more about converse theorems on another occasion.

That side of a triangle which is opposite to any angle, is said to 'subtend' it. Literally, the word 'subtend' means 'to stretch under,' or 'in front of!

EXERCISE VIII.

(The same figure is referred to throughout.)

1. In the figure annexed, given CA equal to CB, name two equal angles.

2. If DC, DB are equal, name another pair of equal angles.

3. If DC is equal to DB, prove that AC cannot be equal to AB.

A D

B

4. If it be certain that, if something called A is like B, then X will be like Y; but, on examining X and Y, we see they are not alike; will A and B be alike or not? 5. If it be certain that, if two magnitudes X and Y are equal, then the triangle CDB in the figure will be as large as the triangle CAB; what may we then say about X and Y?

6. In the figure, looking at ACB and DBC as separate triangles, what side have they in common?

7. It is also given, that DB is equal to AC; now write two sides of the one triangle equal to two of the other.

8. Will this be sufficient to draw any conclusion from by the aid of Prop. IV.? If not, say what more will be required.