PROPOSITION XVII. THEOREM. Any two angles of a triangle are together less than two right angles. GIVEN the triangle ABC; IT IS REQUIRED TO PROVE that any two of its angles are together less than two right angles. B Produce BC to D. Because ACD is an exterior angle of the triangle ABC; Post. 2. Constr. I. 16. Because the angle ACD is greater than the angle ABC, Proved. therefore the angles ACD and ACB are together greater than Because the angles ACD and ACB are together equal to two right angles, and the angles ACD and ACB are together greater than the angles ABC and ACB; W. I. 13. Proved. t. therefore the angles ABC and ACB are together less than two right angles. In the same manner it may be shown that the angles BAC and ACB, as also the angles CAB and ABC are together less than two right angles. Therefore, any two angles &c. Q. E. D. Postulate 2. A terminated straight line may be produced to any length in a straight line. Proposition XVI. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles. Proposition XIII. The angles which one straight line makes with another straight line on one side of it, are together equal to two right angles. The greater side of every triangle has the greater angle opposite to it. In the triangle ABC; GIVEN that AC is greater than AB; IT IS REQUIRED TO PROVE that the angle ABC is greater than the angle ACB. Because ADB is the exterior angle of the triangle BDC; Because AD is equal to AB; therefore the angle ADB is equal to the angle ABD. Because the angle ADB is greater than the angle ACB, Because the angle ABC is greater than the angle ABD, Therefore, the greater side &c. Нур. I. 3. Post. 1. Constr. I. 16. Constr. I. 5. Proved. t. Axiom 9. V. Q. E. D. Proposition III. From the greater of two given straight lines a part may be cut off equal to the less. Postulate 1. A straight line may be drawn from any one point to any other point. Proposition XVI. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles. Proposition V. The angles at the base of an isosceles triangle are equal. Axiom 9. The whole is greater than its part. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it. In the triangle ABC; GIVEN that the angle ABC is greater than the angle ACB; B Because, if AC were equal to AB, the angle ABC would be equal to the angle ACB, and the angle ABC is greater than the angle ACB; Because, if AC were less than AB, the angle ABC would and the angle ABC is greater than the angle ACB; Because AC is not equal to or less than AB; therefore AC is greater than AB. I. 5. Hyp. G. I. 18. G. Proved. 0. Therefore, the greater angle &c. Q. E. D. |