5. An acute-angled triangle is that which has three acute angles.

6. If one side of a triangle be produced, the exterior angle is greater than either of the interior and opposite angles.

7. If one side of a triangle be produced, the exterior angle is equal to the two interior and opposite angles.

8. If each of the sides of a triangle be produced, the three exterior angles are, together, equal to four right angles.

SECTION IV.

TWO TRIANGLES-THEIR EQUALITY.

M.-What may be said, on comparing the angles of two triangles ?

P.-1. The angles of one triangle are, together, equal to the angles of any other triangle; because, their sum, in each, is equal to two right angles.

2. One angle of the one may be equal to an angle of the other.

3. Two angles in the one may be equal to two angles in the other, each to each.

4. The three angles of the one may be equal to the three angles of the other, each to each.

5. The three angles of the one may be unequal to the three angles of the other, each to each.

M.-If an angle of one triangle be equal to an angle of another triangle, what may be said of the other two angles, in each ?

P.—The sum of the other two angles of the one triangle must be equal to the sum of the remaining

two angles of the other; because, the sum of the three angles of the one is equal to the sum of the three angles of the other, and if the equal angles be subtracted from these equals, the remaining angles must be equal.

M.—If two triangles have two angles of the one equal to two angles of the other, each to each, what may be said of the remaining third angles?

P. They must be equal,-for the reason alleged in the former case.

M.-What may be said, then, of the angles of these triangles ? '

P.-The angles of the one are equal to the angles of the other, each to each.

M.-Draw two triangles having one angle of the one equal to one angle of the other.

a bc, so, that the point d may be upon the point a, and the angle e df upon the angle b a c?

P.-The side e d must fall upon the side a b, and

"the side df upon the side a c.

M.-And, where may the points e and ƒ fall?

P.-Either somewhere on the

sides a b and a c,—when de and dƒ are, each, less than a b and a c;

Or,

one of them

may fall upon α b and the

other beyond the point c,-when d e is less than a b, and d f is greater than a c; b

e

a

M.—And, when will the points e and ƒ fall exactly

upon the points b and c?

P.-When d e a b, and d f = a c.

M.-And where, then, will the side eƒ fall?

P.-ef must fall upon b c, and ef must, also, be equal to be; because, since the point e falls upon b, and the point ƒ upon c, the whole line e ƒ must fall upon the whole line b c, and be equal to it,-for, ef and b c are straight, not curved, lines.

M.-And, what may be said of the remaining two angles of each triangle?

P.-The angle d e f must be equal to the angle a b c, and the angle dƒe must be equal to the angle a cb.

M.-And what may, then, be said of the triangles themselves?

P.-The triangle d e f must be equal to the triangle a b c.

M.—Then, when are two triangles equal to each other?

This question the author has been accustomed to propose to his class previously to the preceding investigation, which is strictly in accordance with the demonstration of Euclid. (B. I. Prop. 4). The answers of his pupils were, however, generally of such a nature as to render the demonstration too loose and unmathematical; and he has, accordingly, found it necessary to lead their thoughts into a chain of reasoning similar to the preceding.

P.-Two triangles are equal, when two sides of the one, with the included angle, are equal to two sides of the other, each to each, with the included angle.

"the

M.-Instead of, "the included angle," or angle between them," say, "the angle contained by them." What may be said of the third sides of two such triangles?

P.—Their third sides are equal.

M.-And, of the remaining angles in each?

P. The remaining angles of the one are equal to the remaining angles of the other, each to each.

M.-Could the angle a b c be equal to the angle def?

P.-No: because if the triangle d ef be applied to the triangle a b c so that the point a may be on the point d, and the side ab upon the side, equal to it, df, the side a c will fall upon de,

because bac = Ledf;

the point ƒ will fall upon the point b,

because a b = df;

the point c will fall upon the point e,

because a c=de;

and, hence, the third side b c must fall upon ef, and be, therefore, equal to ef.

Also the triangle a b c must fall upon the triangle def, and be, therefore, equal to it;

upon the

and the abc

Hence, these angles must be equal: and the angle

a b c cannot be equal to the angle de f, unless be likewise equal to dfe.

def

M.-What may be said of the sides which are op

posite to the equal angles, in each triangle?

P.-The sides opposite to the equal angles are equal.