Any two angles of a triangle are together less than two right angles. Let ABC be any ▲. Then must any two of its s be together less than two rt. 4 s. Then Produce BC to D. LACD is greater than ABC. I. 16. :. 48 ACD, ACB are together greater than 48 ABC, ACB. But ¿s ACD, ACB together=two rt. ≤ s. I. 13. .. 48 ABC, ACB are together less than two rt. 4 s. Similarly it may be shewn that s ABC, BAC and also that s BAC, ACB are together less than two rt. 2 s. NOTE 4. On the Sixth Postulate. Q. E. D. We learn from Prop. xvII. that if two straight lines BM and CN, which meet in A, are met by another straight line DE in the points O, P, B M A the angles MOP and NPO are together less than two right angles. The Sixth Postulate asserts that if a line DE meeting two other lines BM, CN makes MOP, NPO, the two înterior angles on the same side of it, together less than two right angles, BM and CN shall meet if produced on the same side of DE on which are the angles MOP and NPO. PROPOSITION XVIII. THEOREM. If one side of a triangle be greater than a second, the angle opposite the first must be greater than that opposite the second. B In ▲ ABC, let side AC be greater than AB. Then must ABC be greater than ▲ ACB. From AC cut off AD=AB, and join BD. I. 3. Then ... AB=AD, .. ▲ ADB= ▲ ABD, I. A. And CD, a side of ▲ BDC, is produced to A. I. 16. :. ADB is greater than 4 ACB ; ..also 4 ABD is greater than ▲ ACB. Much more is ABC greater than ▲ ACB. Q. E. D. Ex. Shew that if two angles of a triangle be equal, the sides which subtend them are equal also (Eucl. I. 6). PROPOSITION XIX. THEOREM. If one angle of a triangle be greater than a second, the side opposite the first must be greater than that opposite the second. B In ▲ ABC, let 4 ABC be greater than 4 ACB. For if AC be not greater than AB, AC must either=AB, or be less than AB. Now AC cannot=AB, for then I. A. ▲ ABC would = ▲ ACB, which is not the case. And AC cannot be less than AB, for then I. 18. ▲ ABC would be less than ACB, which is not the case; .. AC is greater than AB. Q. E. D. Ex. 1. In an obtuse-angled triangle, the greatest side is opposite the obtuse angle. Ex. 2. BC, the base of an isosceles triangle AC, is produced to any point D; shew that AD is greater than AB. Ex. 3. The perpendicular is the shortest straight line, which can be drawn from a given point to a given straight line; and of others, that which is nearer to the perpendicular is less than the more remote. Any two sides of a triangle are together greater than the third side. B Let ABC be a ▲. Then any two of its sides must be together greater than the third side. Produce BA to D, making AD=AC, and join DC. that is, BD=BA and AC together; .. BA and AC together are greater than BC. Similarly it may be shewn that AB and BC together are greater than AC, and BC and CA AB. I. A. I. 19. Q. E. D. Ex. 1. Prove that any three sides of a quadrilateral figure are together greater than the fourth side. Ex. 2. Shew that any side of a triangle is greater than the difference between the other two sides. Ex. 3. Prove that the sum of the distances of any point from the angular points of a quadrilateral is greater than half the perimeter of the quadrilateral. Ex. 4. If one side of a triangle be bisected, the sum of the two other sides shall be more than double of the line joining the vertex and the point of bisection. S. E. If, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle; these will be together less than the other sides of the triangle, but will contain a greater angle. E Let ABC be a ▲, and from D, a pt. in the ▲, draw st. lines to B and C. Then will BD, DC together be less than BA, AC, Produce BD to meet AC in E. Then BA, AE are together greater than BE. I. 20. Add to each EC. Then BA, AC are together greater than BE, EC. Again, DE, EC are together greater than DC. I. 20. Add to each BD. Then BE, EC are together greater than BD, DC. And it has been shewn that BA, AC are together greater than BE, EC; Next, .. BA, AC are together greater than BD, DC. Ex. 1. Upon the base AB of a triangle ABC is described a quadrilateral figure ADEB, which is entirely within the triangle. Shew that the sides AC, CB of the triangle are together greater than the sides AD, DE, EB of the quadrilateral. |