In the triangle ABC;

GIVEN AB equal to 4C,

IT IS REQUIRED TO PROVE the angle ABC equal to the angle ACB.

A

B'

Imagine the triangle ABC in another position as A'B'C'.

Apply the triangle ABC to the triangle A'B'C' so that A may be on A', AB on A'C' and C on the same side of A'C' as B'.

Нур.

Because AB is equal to A'C',

therefore B is on C'.

Because the angle BAC is the same as B'A'C',

therefore AC is on A'B'.

Because AC is equal to A'B',

therefore C is on B'.

Because B is on C" and C on B',

and two straight lines cannot enclose a space,

'therefore BC coincides with B'C'

and the angle ABC is equal to the angle A'C'B', that is

to ACB.

Нур

Proved.

Axiom.

In the triangle ABC;

GIVEN the angle ABC equal to the angle ACB;
IT IS REQUIRED TO PROVE AB equal to AC.

B'

Imagine the triangle ABC in another position as A'B'C'.

Apply the triangle ABC to the triangle A'B'C' so that B may be on C', BC on C'B' and A on the same side of B'C' as A'.

Because BC is the same as C'B',

therefore C is on B'.

Because the angle ABC is equal to the angle A'C'B',

therefore A is in C'A'.

Because the angle ACB is equal to the angle A'B'C',

therefore A is in A'B'.

Because A is in both A'B' and A'C',

Hyp.

Нур.

Proved.

therefore A is on A',

and AB is equal to A'C',

that is to AC.

In the triangles ABC and DEF;

GIVEN that AB is less than DE, AC less than DF, and the angle BAC equal to the angle EDF;

IT IS REQUIRED TO PROVE that the triangle ABC is less than the triangle DEF.

Apply the triangle ABC to the triangle DEF so that A is on D, AB on DE and C on the same side of DE as F.

Because AB is less than DE,
therefore B is in DE.

Because the angle BAC is equal to the angle EDF,

therefore AC is on DF.

Because AC is less than DF,

therefore C is in DF,

Нур.

Hyp.

Hyp.

and the triangle ABC is less than the triangle DEF. Axiom 9.

In the triangle ABC;

GIVEN that BD and CE the bisectors of the angles ABC and ACB are equal,

IT IS REQUIRED TO PROVE that the angles ABC and ACB are equal.

B

E

B/

A A

Imagine the triangle ABC in another position as A'B'C'. Apply the triangle ABC to the triangle A'B'C' so that C may be on B' and CE on B'D'.

Because CE is equal to B'D',

therefore E falls on D'.

Нур.

Supposing the angle ACB not equal to A'B'C', let ACB be the greater, Because the angle ACB is greater than the angle A'B'C' Supposed. and each is bisected by B'D',

therefore B'A and B'B each falls without the angle A'B'C'.

Join AA'.

In the triangles AOB' and A'OD' ;

Because the angle AOB' is equal to the angle A'OD', and the angle OAB' is the same as the angle OA'D', therefore the angle AB'O is equal to the angle A'D'O. In the triangles AB'A' and AD'A';

Hyp.

I. 15.

I. 32.

Because AB' is greater than its part A'D',
and A'B' is greater than its part AD',

Axiom 9.

Axiom 9.

Proved.

and the angle AB'A' is equal to the angle AD'A',

therefore the triangle AB'A' is greater than the triangle AD'A'. Page 55. Take away from each the triangle AOA' and add the quadrilateral

OD'C'B',

therefore the quadrilateral AD'C'B' is greater than the

triangle A'B'C'.

Axioms 2 and 3.

Much more then is the triangle AB'B greater than A'B'C',

which is impossible.

Therefore the angle ACB is equal to the angle ABC.

MISCELLANEOUS QUESTIONS.

(164) Define the terms magnitude, space magnitude, postulate, axiom, proposition, theorem, problem, hypothesis, construction, demonstration, conclusion, corollary.

(165) Define the terms straight or right line, segment of a line, finite line, radius of a circle, diagonal, perimeter, hypotenuse, altitude of a triangle, base of a triangle, vertex of a triangle, point of section, an angle, equal angles, vertical angles, exterior or external angles, interior angles, adjacent angles, opposite angles, alternate angles, equal triangles, equivalent triangles, plane figure, regular figure, area, equilateral figure, equiangular figure, rectangular figure, rectangle, complements, bisector of an angle.

(166) Name the (a) one-sided, (b) two-sided, (c) three-sided, (d) foursided figures defined by Euclid; and give the derivations of the names given by him to all plane figures having more than four sides. What does he call quadrilateral figures which are not parallelograms?

(167) Name all the space magnitudes treated of by Euclid in his first book, and give examples of magnitudes that are not space magnitudes.

(168) Which of the axioms are true of all magnitudes, and which of space magnitudes only?

(169) Of how many dimensions of space does Euclid treat in his first book?

Name them.

(170) Which of the thirty-six definitions given, are not required

in the first book?

(171) Name the postulates which are assumed, and the problems which are proved in the first book.

(172) Distinguish between the converse and the contrary of a theorem; and also between direct and indirect demonstrations.

(173) State all the cases in Euclid's first book of a proposition and its converse being proved (a) one directly and one indirectly, (b) both directly, (c) both indirectly. Name also those propositions whose converses are true, but are not demonstrated. Are there any propositions whose converses are not true?

(174) In order to demonstrate his last two propositions, Euclid requires twenty-six of his former propositions; which are they?