much more.. Lbac > Labc.

Hence, in every triangle the greater angle is opposite to the greater side.

M.—If, then, it be known that one angle of a triangle is greater than another angle, what must be concluded with respect to the sides subtending [opposite to] these angles?

P.-The side subtending the greater angle must be greater than the side subtending the less.

M.—Which, then, of the sides of a right-angled triangle is the greatest?

P.-The side subtending the right angle.

M.—And in an obtuse-angled triangle, which is the greatest side?

P. The side subtending the obtuse angle.

M. In an equilateral triangle, compare the sum of any two sides with the remaining side.

P.-Any two sides of an equilateral triangle must together be greater than the remaining [third] side, because all the sides are equal.

M.-But, in any triangle, are two sides together greater or less than the remaining third side?

P.-The sides, b a + a c must be

greater than b c, because b c is the

shortest distance between the points

b and c.

α

b

For same reason, ab + bc>a c, and ac + bc> ab.

M.-This truth may be demonstrated by converting two sides into one: endeavour to do so. P.-Produce the side a bat the

point a, and make a d, the part produced, equal to a c, and join dc:

b.

α

And to the greater angle the greater side is opposite; de,

.. bd

but bdba + ac:

.ba + ac>bc.

Hence, any two sides of a triangle are together greater than the remaining side.

M. In the triangle a b c, let

b

a

cess of b c over a b―i. e. by how much bc is greater than a b?

P.-By cutting off from the greater, bc, a part equal to ab; the remaining part must be the difference, in length, between b c and a b.

M.-Compare this difference between two unequal sides of a triangle with the remaining [third] side.

P.-Let be be greater than a b

from b c cut off b d = a b :

;

b

de is the difference between bc and ab.

a

d

Then, because any two sides of a triangle are to

gether greater than the third side,

ba + ac>bc;

but ba=bd;

From these unequals take-away bd, which is common

to both

there remains ac dc.

Hence, the difference between any two sides of a triangle is less than the third side.

M.-The same truth can be démonstrated by means of the angles try this method.

P. Let d c be the difference

between b c and ab; join ad: then · ba = b d,

▲ bad=bda;

b

but the ext.ade> int. opp. bad;

..Lade> Lbda.

Also, the ext. bda> int. opp. da c;

much more.. Lade> <dac;

but to the greater angle the greater side is opposite ...ac>dc

that is, the difference, dc, between b c and a b, is less than the third side, a c.

M.-Compare the three sides of a triangle with the double of any one side.

P. The three sides of any triangle are together greater than double the length of any one side;

for, abac being > bc,

add be to each of these unequals;

then abac + bc>bc + bc.

b

a

SUBSTANCE OF SECTION IV.

1. If two triangles have one angle of the one equal to one angle of the other, the sum of the remaining

two angles of the one is equal to the sum of the remaining two angles of the other.

2. If two triangles have two angles of the one equal to two angles of the other, each to each, the third angle of the one is equal to the third angle of the other; that is, the triangles are equiangular.

3. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by these sides equal, their third sides are equal, the triangles are equal, and the remaining angles of the one are equal to the remaining angles of the other, each to each, namely, those to which the equal sides are opposite.

4. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of the one greater than the angle contained by the two sides, equal to them, of the other, the base of that triangle which has the greater angle is greater than the base of the other.

5. The angles at the base of an isosceles triangle are equal.

6. In an isosceles triangle, the straight line which bisects the vertical angle bisects the base.

7. In an isosceles triangle, the straight line which bisects the vertical angle stands at right angles to [is perpendicular to,] the base.

8. In an isosceles triangle, if the base be bisected, the straight line joining the vertical angle and the point of bisection bisects the vertical angle and stands at right angles [is perpendicular to,] to the base.

9. If a triangle be equilateral, it is likewise equiangular.

10. If two angles of a triangle be equal to one another, the sides opposite to them are likewise equal; that is, the triangle is isosceles.

11. If a triangle be equiangular, it is likewise equilateral.

12. In every triangle the greater side is opposite to the greater angle.

13. In every triangle the greater angle is subtended by the greater side.

14. Any two sides of a triangle are together greater than the third side.

15. The difference between any two sides of a triangle is less than the third side.

16. The three sides of every triangle are together greater than double the length of any one side.

SECTION IV.

EQUALITY OF TRIANGLES.

M.-Describe a triangle; in it take any point, and, from the extremities of any one side of the triangle, draw lines to that point: what is the result?

P.-Two triangles, a b c

d

and b d c.

b

a

M.-What have these two triangles in common?

P.-The base b C.