then shall any two of its angles be together less than two right angles.

Produce BC to D.

Because ACD is an exterior angle of the triangle ABC, therefore it is greater than the interior opposite angle ABC, I. 9. to each add the angle ACB,

then the angles ACD, ACB are together greater than the angles АВС, АСВ;

Ax. f.

but the angles ACD, ACB are together equal to two right angles, I. 2.

therefore the angles ABC, ACB are together less than two right angles.

In the same way it may be shown that the angles BCA, CAB, or the angles CAB, ABC are together less than two right angles.

Q.E.D.

COR. 1. If a triangle has one right angle or obtuse angle, its remaining angles are acute.

COR. 2. Every triangle has at least two acute angles.

Hence,

DEF. 30. A triangle which has one of its angles a right angle is called a right-angled triangle. A triangle which has one of its angles an obtuse angle is called an obtuse-angled triangle. A triangle which has all its angles acute is called an acute-angled triangle.

DEF. 31. The side of a right-angled triangle which is opposite to the right-angle is called the hypotenuse.

COR. 3. From a given point outside a given straight line, only one perpendicular can be drawn to that line.

Ex. 14. Prove Theor. 10 by joining the vertex to any point in the base, and using Theor. 9 twice.

THEOR. 11. If two sides of a triangle are unequal, the greater side has the greater angle opposite to it.

Let ABC be a triangle having the side AB greater than the side AC:

B

A

then shall the angle ACB be greater than the angle ABC.

From AB cut off AD equal to AC, join CD.

Then because AD is equal to AC,

therefore the angle ADC is equal to the angle ACD ;

I. 7.

but the angle ADC, being an exterior angle of the triangle BDC, is greater than the interior opposite angle ABC,

therefore the angle ACD is greater than the angle ABC; still more then is the angle ACB greater than the angle ABC.

Ex. 15. Prove Theor. 8 by means of Theor. II.

I. 9.

Q.E.D.

D

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; I. 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,

I. 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;

I. 12.

but BA and AC are together equal to BD,

since AC is equal to AD,

therefore BA and AC are together greater than BC.

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 :

E

D

B

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,

1. 13.

then BA and AC are together greater than BE and EC; Ax. f.