Because if BD does not coincide in direction with BE, and this is impossible; therefore BD does coincide in direction with BE, Therefore, if at a point &c. Proved. Axiom 9. G. Q. E. D. Postulate 2. A finite straight line may be produced to any length in a straight line. 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. Axiom 1. Magnitudes which are equal to the same magnitude are equal to one another. Axiom 3. If equals be taken from equals the remainders are equal. Axiom 9. The whole is greater than its part. Axiom 11. All right angles are equal to one another. Axiom 6. Magnitudes which are double of the same magnitude are equal to one another. PROPOSITION XV. THEOREM. If two straight lines cut one another, the vertical, or opposite angles shall be equal. GIVEN that the two straight lines CD and AB cut one another at the point E; IT IS REQUIRED TO PROVE that the angle AEC is equal to the angle BED, and also that the angle AED is equal to the angle BEC. therefore the angles AEC and AED are together equal to two right angles. Because DE falls on AB; therefore the angles AED and BED are together equal to two right angles. Because the angles AEC and AED are together equal to two right angles, and the angles AED and BED are together equal to two right angles; Hyp. I. 13. Hyp. I. 13. Proved. Proved. therefore the angles AEC and AED are together equal to the angles AED and BED. Axioms 11, 6 and 1. Because the angles AEC and AED are together equal to the angles AED and BED; Proved. therefore the angle AEC is equal to the angle BED. Axiom 3. In the same manner it may be shown that the angle AED Therefore, if two straight lines cut one another &c. Q. E. D. Proposition XIII. The angles which one straight line makes with another on one side of it are together equal to two right angles. Axiom 1. Magnitudes which are equal to the same magnitude are equal to one another. Axiom 3. If equals be taken from equals the remainders are equal. Axiom 11. All right angles are equal to one another. Axiom 6. Magnitudes which are double of the same magnitude are equal to one another. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles. In the triangle ABC; GIVEN that BC is produced to D; IT IS REQUIRED TO PROVE that the angle ACD is greater than either of the angles CBA and BAC. and the angle AEB is equal to the angle FEC; therefore the angle BAE is equal to the angle ECF. Because the angle ACD is greater than the angle ECF, I. 10. Post. 1. Post. 2. I. 3. Post. 1. Constr. Constr. I. 15. I. 4. Axiom 9. t. In the same manner, if BC is bisected and AC produced to G, it may be proved that the angle BCG (which is equal to the angle ACD) is greater than the angle ABC. Therefore, if one side of a triangle &c. I. 15. Q. E. D. Proposition X. A given finite straight line may be bisected. Postulate 1. A straight line may be drawn from any one point to any other point. Postulate 2. A terminated straight line may be produced to any length in a straight line. Proposition III. From the greater of two given straight lines a part may be cut off equal to the less. Proposition XV. If two straight lines cut one another the vertical or opposite angles are equal. Proposition IV. If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the included angles equal, the other angles are equal each to each, namely those to which the equal sides are opposite. Axiom 9. The whole is greater than its part. |