An elementary treatise on geometry : simplified for beginners not versed in algebra . at the vertex. (Truth I.) Q. But of what use is your proving that the triangleORI is equal to the triangle OPF ? A. It shows that since the triangle OPF is right-an-gled in P, the triangle ORI must be right-angled in R;for, in equal triangles, the equal angles are opposite tothe equal sides (remarks to Query 6, page 25); conse-quently the two lines AB, CD, are both perpendicular tothe same straight line PR, and therefore parallel to eachother. (Last query.) Q. Supposing, now, twostraight lines, AB, CD, tobe c
An elementary treatise on geometry : simplified for beginners not versed in algebra . at the vertex. (Truth I.) Q. But of what use is your proving that the triangleORI is equal to the triangle OPF ? A. It shows that since the triangle OPF is right-an-gled in P, the triangle ORI must be right-angled in R;for, in equal triangles, the equal angles are opposite tothe equal sides (remarks to Query 6, page 25); conse-quently the two lines AB, CD, are both perpendicular tothe same straight line PR, and therefore parallel to eachother. (Last query.) Q. Supposing, now, twostraight lines, AB, CD, tobe cut by a third line, MN, soas to make the alternate an-gles AEF and EFD, or theangles BEFand EFC, equal,what relation would the linesAB, CD, then bear to each other ? A. TJiey vwuld still be parallel, Q. How can you prove this 1 A. If the angle AEF is equal to the angle EFD, theangles AEF and CFN are also equal; because EFD andCFN are opposite angles at the vertex. And, in the samemanner, it may be proved, that if the angles BEF andEFC are equal, MEA and EFC are also equal; there-. GEOAtETRY. 29 fore, in both cases, there are two straight lines cut by athird line at equal angles; consequently they are parallelto each other. Q. There is one more case, and that is : If the twostraight lines AB, CD (in our last figure), are ait hy athird line MN, so as to make the sum of the two interiorangles AEF and EFC, equal to tioo right angles, howare the straight lines AB, CD, then, situated with regardto each other ? A. They are still parallel to each other. For the sumof the two adjacent angles EFC and CFN is also equalto two right angles ; and therefore, by taking from eachof the equal sums the common angle EFC, the two re-maining angles AEF and CFN must be equal (TruthIV.); and you have again the first case, viz : two straightlines cut by a third line at equal angles. Q. Will you now state the different cases in which twostraight lines are parallel ? A. 1. Tllien they are cut by a third
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