hypothesis from conclusion in the enunciation of Prop. VII. 6. State the nature of the proof of the same proposition. 7. What is the purpose of using the word “finite” in the enunciation of Prop. X.? 8. Analyse Prop. XI. into datum, quaesitum, particular enunciation, construction, and proof. 9. In the figure of Prop. XI., suppose the triangle DEF were made isosceles, what alteration would be required in the proof? 10. In the construction of XII., suppose D taken so far beyond 4B that the point A falls within the circle EFG; how would that affect the construction ? 11. Two lines are drawn in the same which are not necessary for obtaining the perpendicular. Name them, and say why they are drawn. 12. Define a chord of a circle. SECTION III. PROPOSITIONS XIII.-XVII. On the Angles which Two Intersecting Straight Lines make with one another or with a Third. 46. EXERCISE XVIII. 1. Name four pairs of adjacent angles made с at the point E by the straight lines AB, CD. 2. Which of these pairs of adjacent angles is on one side of AB and above it? Which is E В below the same line? Which is on the same ID side of CD as B? 3. If AEC and BEC be equal, what may we call each of them ? (Quote the definition which applies.) 4. If AEC, BEC be not equal, and a straight line be drawn upwards from E at right angles to AB, show that it cannot coincide with EC. 5. Hence infer that, if AEC, BEC be unequal, one is acute, the other obtuse, 6. In the second figure to Prop. XIII., CBE is a right angle. By how much is the angle CB A greater than a right angle? How much is the angle DB A less than a right angle? 7. In the same figure, take the angle EBA from the angle CBA and add it to the angle ABD; what two angles have thus been constructed out of the two angles CBA, ABD? Draw an inference with regard to the magnitude of the sum of any two adjacent angles. 8. Take the three angles of the same figure, and show that by adding two of them together we may form two right angles out of them. 9. Show that the two angles CBA, ABD may be constructed out of the same three angles. 10. By means of the two constructions of Questions 8 and 9, show that the angles CBA, ABD are together equal to two right angles. 47. PROPOSITION XIII.—THEOREM. The angles which one straight line makes with another, upon one side of it, are either two right angles, or are together equal to two right angles. Let the straight line AB make with CD, upon one side of CD, the angles CBA, ABD. Then these shall either be two right angles, or shall be together equal to two right angles. For, if the angle CBA be equal to the angle ABD, each of them is a right angle. (Def. 10) But, if the angles CBA, ABD be not equal, from the point B draw BE at right angles to CD; (I. 11) so that the angles CBE, EBD are two right angles. The whole angle EBD is equal to the two EBA, ABD; add the angle CBE to each of these equals; then the angles CBE, EBD are equal to the three angles CBE, EBA, ABD. (Ax. 2) Again, the angle CBA is equal to the two CBE, EBA; to each of these equals add the angle ABD; then the angles CBA, ABD are equal to the three angles CBE, EBA, ABD. But the angles CBE, EBD have been proved equal to the same three angles; and things which are equal to the same are equal to one another; therefore the angles CBA, ABD are equal to the angles CBE, EBD. (Ax. 1) But CBE, EBD are two right angles. (Constr.) Therefore the angles CBA, ABD are together equal to two right angles. Wherefore the angles which one straight line, &c. Q. E. D. 48. SUPPLEMENTARY ANGLES. Def.—When two angles, taken together, are equal to two right angles, either of them is called the supplement of the other. Such angles are called supplementary.' Proposition XIII. treats of the most important case of supplementary angles ; namely, those which are adjacent. It may be observed that Prop. XIII. is proved on the simplest principles we can employ ; namely, the earlier axioms. There is nothing in its proof, therefore, to prevent its being made the first proposition in the subject. The construction, however, will supply the reason for its present position (after Prop. XI.); for Euclid never adds a line of construction whose method has not previously been established. Some writers on Geometry consider Euclid's caution in this regard to be carried farther than is necessary for safe argument. EXERCISE XIX. 1. In the application of Axiom 1, in the proof, state the magnitude which is used for successive comparison with two others. Also state distinctly these other magnitudes. 2. In the figure of Ex. XVIII. 1, divide the angles at E into two pairs of adjacent angles ; and deduce from Prop XIII. the magnitude of all the four taken together. 3. In the figure of Prop. XIV., if CB, BE form one straight line, what may the angles CBA, ABE be called ? 4. Given, in the figure to Prop. XV., that AB, CD are two straight lines; state the magnitude of the two angles CEA, AED taken together. 5. Show that the sum of those angles is equal to the sum of the angles BED, DEA. 6. Applying Axiom 3 to these equals, show that two of the angles mentioned are equal. 7. By a similar method, show that the angles AED, BEC are equal. 8. If two straight lines intersect, show that two adjacent angles thus formed cannot be both acute, nor can they be both obtuse. 49. PROPOSITION XIV.-THEOREM. If at a point in a straight line two other straight lines upon the opposite sides of it make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line. At the point B in the straight line AB, let the straight lines BC, BD, upon opposite sides of AB, make the adjacent angles ABC, ABD together equal to two right angles. Then BD shall be in the same straight line with BC. For, if BD be not in one straight line with BC, let BE be in a straight line with BC, Then, because CBE is one straight line, therefore the angles CBA, ABE are together equal to two right angles ; (1.13) but the angles CBA, ABD are equal to two rightangles; (Hyp.) therefore CBA, ABE are together equal to CBA, ABD. Take the common angle CBA from these equals; then the remaining angles ABE, ABD are equal ; (Ax. 3) the less equal to the greater; which is impossible. Therefore BE is not in the same straight line with BC. it be demonstrated that Q. E. D. 50. PROPOSITION XV.-THEOREM. If two straight lines cut one another, the opposite vertical angles shall be equal. Let the two straight lines AB, CD cut one another in the point E. Then the angle AEC shall be equal to the angle BED, and the angle AED to the angle BEC. Because AE makes with CD the adjacent angles CEA, AED, these angles are together equal to two right angles. (I. 13) Again, because DE makes with AB the adjacent angles AED, DEB, these angles also are together equal to two right angles. (I. 13) But CEA, AED have been shown to be together equal to two right angles. Therefore CEA, AED are together equal to AED, DEB. Take away the common angle AED; then the remaining angle CEA is equal to the remaining angle DEB. (Ax. 3) In the same manner, it may be demonstrated that the angle AED is equal to the angle BEC. Wherefore, if two straight lines cut one another, &c. Q. E. D. Cor. 1.–From the mode of demonstration, it is manifest that, if two straight lines cut each other, the angles which they make at the point where they cut are together equal to four right angles. Cor. 2.—And, consequently, that all the angles made by any number of straight lines, meeting in one point, are together equal to four right angles. Note. These two important corollaries are usually given here; but |