Therefore, also, the sum of the three interior 28 of the AA B C is equal to two right 8. In the same way it may be proved that the sum of the three interior 8 of any other is equal to two right 8. This important proposition involves an immense variety of consequences. Some of the most obvious are stated in the following corollaries : 1. If one of a A is a right Z, the other two <s are together equal to a right Z. 2. If one of a mis equal to the sum of the other two <s, that is a right Z. 3. An obtuse is greater, and an acute is less than the sum of the other two 28. 4. If one of a A is greater than the sum of the other two, it is obtuse; if it is less than the sum of the other two, it is acute. 5. If two isosceles As have their vertical Zs equal, the Zs at their bases are also equal. 6. If the vertical L of an isosceles A is a right <, each of the 28 at the base is half a right L. 7. The sum of all the internal 8 of any recti linear figure, together with four right Z8, is equal to twice as many right s as the figure has sides. Let ABCD E be a rectilinear figure. Take a point F somewhere inside the figure, and join the points F and A F and B, F and C, F and D, F and E The whole figure is thus divided into as many as as the figure has sides. The sum of the three xs of each of these si two right <8; consequently, the sum of all the <8 of all the As is twice as many right s as the figure has sides. The 4s at the point F together make up four B right <8; the other 8 of the A8 make up the 8 of the polygon AB CD E. Therefore the <s of the polygon together with four right Z8 are equal to twice as many right Z8 as the figure has sides. 8. If the sides of any convex rectilinear figure be produced, the external Zs are together equal to four right _8. For each external < with the internal 2 adjacent to it, makes up the two right Zs. Consequently, all the internal <s and all the external s taken together make up twice as many right 28 as the figure has sides. But all the internal <s, together with four right <s, make up twice as many right 8 as the figure has sides. It follows, therefore, that all the external Zs taken together, make up four right Zs. 9. Each of an equilateral A is the third part of two right 8, or two-thirds of one right Z. Therefore, if we bisect an % of an equilateral A, we get the third part of a right Z. (See note on Prop. IX.) PROPOSITION XXXIII. If there are two straight lines which are equal and parallel to each other, and their corresponding* extremities be joined by straight lines, these straight lines will also be equal and parallel. For the construction in this proposition we must be able to join two given points by a straight line. * By corresponding extremities are meant those which point in the same direction. To prove the proposition by the help of the con struction we must know,1. That if two As have two sides of the one equal respectively to two sides of the other, and also the < included between the said sides in the first A equal to the < included between the said sides in the other A, those As will also be equal to each other in every other respect. (Prop. IV.) 2. That if two straight lines are parallel and a third line meets them, the alternate Z8 so formed are equal. (Prop. XXIX.) 3. That if the alternate <s, formed when two straight lines are met by a third, are equal, those two straight lines will be parallel. (Prop. XXVII.) Suppose CD and A B to be two equal and parallel straight lines, and let the corresponding extremities C and A and D and B be joined by the straight lines CA and D B. It has to be proved that the lines CA and D B are equal and parallel. Join by a straight line either pair of extremities of the equal and parallel straight lines, which do not correspond, as, for example, C and B. We have thus formed two As, CD B and BAC. In these As the side CD is equal to the side B A. (This we know to start with.) The side C B is common to the two As. Also the Zs D C B and C B A are equal, because they are the alternate <s formed by the parallel straight lines C D and A B when met by a third lino, U B. Thus the two As CD B and BA O have two sides of the one (namely D C and C B) equal respectively D A B to two sides of the other (namely A B and B C), and the included <DCB of the one equal to the included < ABC of the other. Consequently (according to the fourth proposition), these triangles are also equal in every other respect. Among these respects are : 1. That the side B D is equal to the side A C. (This is one part of the proposition to be proved.) 2. That the < D B C is equal to the < ACB. Now D BC and ACB are the alternate <s formed by the two straight lines A C and BD when met by a third line, C B. Hence, since these alternate Zs are equal, the lines AC and B D are parallel. (Prop. XXVII.) It has thus been shown that the lines A C and BD are both equal and parallel. PROPOSITION XXXIV. The opposite sides and angles of a parallelogram are equal to each other, and a parallelogram is bisected (that is, divided into two equal parts) by each of its diagonals. To prove this proposition we must know,- equal. (Ax. II.) between them in the one, equal respectively to two sides and the between them in the other, those As are also equal in every other respect. (Prop. IV.) 3. That if two As have two <s of the one equal respectively to two Zs of the other, and a side of the first A equal to a side of the A с D second, similarly placed with respect to the equal <8, those As are also equal in every other respect. (Prop. XXVI.) 4. That if two lines are parallel and a third line meets them, the alternate <8 so formed are equal. (Prop. XXIX.) Lot ABDC be a parallelogram, that is, suppose it to be a four-sided figure, whose B opposite sides are parallel, and let A D be one of its diagonals. We have to prove that the opposite sides are equal :-namely, the side A B to the side CD, and the side AC to the side BD: and also that the opposite <s are eqnal, namely, the ABD to the < ACD, and the < CAB to the < BDO and that the parallelogram is divided into two equal parts by its diagonal A D. Proof. Because A B and C D are parallel, and the line A D meets them, it follows that the alternate 8 so formed, namely, BAD and ADC, are equal. (Prop. XXIX.) Because A C and B D are parallel and the line A D meets them, it follows (Prop. XXIX.) that the alternate s so: formed, namely, CAD and AD B, are equal. Hence, the two A8 ABD and CDA have the two Łs BAD and BDA of the one equal respectively to the two Zs ADC and CAD of the other, and the side A D common to the two As and similarly placed with respect to the equal 8. It follows, therefore (Prop. XXVI.), that these As are also equal in every other respect. That is to say, the < ABD is equal to the < ACD; the side A B to the side CD, the side B D to the side A C, and the area of the one A to the area of the other. |