THEOR. 12. If two angles of a triangle are unequal, the greater angle has the greater side opposite to it.

Let ABC be a triangle having the angle ABC greater than the angle ACB:

then shall the side AC be greater than the side AB.

If AC is not greater than AB,

then it is either equal to, or less than, AB.

But AC is not equal to AB,

for then the angle ABC would be equal to the angle ACB; 1. 7. also AC is not less than AB,

for then the angle ABC would be less than the angle ACB; I. 11. therefore AC is greater than AB.

Q.E.D.

NOTE. As to this proof and that of Ex. 15 cf. Introduction,

$9.

Ex. 16. The hypotenuse of a right-angled triangle is greater than either of the remaining sides.

Ex. 17. In an obtuse-angled triangle the side opposite the obtuse angle is the greatest.

THEOR. 13. Any two sides of a triangle are together greater than the third side.

Let ABC be a triangle:

then shall the sides BA and AC be together greater than the side BC, AC and CB than AB, and CB and BA than AC.

Produce BA to D, make AD equal to AC, and join CD,
Because AD is equal to AC,

therefore the angle ACD is equal to the angle ADC;

but the angle BCD is greater than the angle ACD,

1. 7.

Ax. a.

therefore the angle BCD is also greater than the angle ADC, that is, than the angle BDC,

therefore the side BD of the triangle BDC is greater than the

side BC;

but BA and AC are together equal to BD,

since AC is equal to AD,

therefore BA and AC are together greater than BC.

I. 12.

Similarly it may be shown that AC and CB are together

greater than AB, and CB and BA than AC.

Q.E.D.

COR. The difference of any two sides of a triangle is less than

the third side.

*Ex. 18. The straight line drawn from the vertex of a triangle to the middle point of the base is less than half the sum Use the construction of

of the remaining sides.

Theor. 9.

Ex. 19. If O is a point within the triangle ABC, shew that the sum of OA, OB, and OC is greater than half the perimeter of the triangle.

Ex. 20. The perimeter of a quadrilateral is greater than the sum, and less than twice the sum of the diagonals.

THEOR. 14. If from the ends of a side of a triangle two straight lines are drawn to a point within the triangle, these are together less than the two other sides of the triangle, but contain a greater angle.

Let ABC be a triangle, and from the ends of a side BC let straight lines BD, CD be drawn to a point D within the triangle:

then shall BD and DC be together less than BA and AC, but the angle BDC shall be greater than the angle BAC. Produce BD to meet AC at E.

Then BA and AE are together greater than BE,

to each of these add EC,

then BA and AC are together greater than BE and EC

I. 13.

Ax. f.

again, DE and EC are together greater than DC,

to each of these add DB,

I. 13.

Ax. f.

then BE and EC are together greater than BD and DC; but BA and AC are together greater than BE and EC, still more then are BA and AC together greater than BD and DC.

Again, the angle BDC, being an exterior angle of the triangle CED, is greater than the interior opposite angle DEC;

I. 9.

and the angle DEC, being an exterior angle of the triangle BAE, is greater than the angle BAC,

still more then is the angle BDC greater than the angle BAC.

Q.E.D. Ex. 21. If a point O be taken within the triangle ABC, the sum of OA, OB, and OC is less than the perimeter of the triangle.

THEOR. 15. Of all the straight lines that can be drawn to a given straight line from a given point outside it, the perpendicular is the shortest; and of the others, those which make equal angles with the perpendicular are equal; and that which makes a greater angle with the perpendicular is greater than that which makes a less angle.

Let A be the given point, and BC the given straight line, and let AD be the perpendicular from A to BC,

and let AE, AF be any straight lines making equal angles EAD, FAD with AD,

and let AG make with AD the angle GAD greater than the angle EAD, or FAD:

the angles ADE and AED are less than two right angles; I. 10.

the angle EAD is equal to the angle FAD,

Нур.

and the side AD is common to both,

1. 6.

therefore AE is equal to AF.

Also, of AE and AF let AE be the one that is on the same

side of the perpendicular as AG,

then, in the triangle AGE,

the angle AEG, being an exterior angle of the triangle AED, greater than the interior opposite angle ADE,

is

I. 9.