Elements of geometry and trigonometry . Y /o D the feum of the interior angles ()(îll, (iOD, woul<i be C(|ual totwo right angles (Prop. W.), whereas it is Irss by hypothesis:iienee, the line-s III, CD, an* not parallel, and will thoreforemeet if suflicienfb »r(,riii<r(l. 28 GEOMETRY. Cor, It is evident that the two lines IH, CD, will meet orthat side of EF on which the sum of the two angles OGH,GOD, is less than two right angles. PROPOSITION XXII. THEOREM. Two straight lines which are parallel to a third line, are parallel to each other. Let CD and AB be parallel to the third line EF ; t
Elements of geometry and trigonometry . Y /o D the feum of the interior angles ()(îll, (iOD, woul<i be C(|ual totwo right angles (Prop. W.), whereas it is Irss by hypothesis:iienee, the line-s III, CD, an* not parallel, and will thoreforemeet if suflicienfb »r(,riii<r(l. 28 GEOMETRY. Cor, It is evident that the two lines IH, CD, will meet orthat side of EF on which the sum of the two angles OGH,GOD, is less than two right angles. PROPOSITION XXII. THEOREM. Two straight lines which are parallel to a third line, are parallel to each other. Let CD and AB be parallel to the third line EF ; then arcthey parallel to each other. Draw PQR perpendicular to EF, and cutting AB, CD. Since AB is parallel to EF, PR will be perpendicular to AB ( XX, Cor. 1.) ; and since CD is parallel to EF, PR will for a like reason be perpen- C dicular to CD. Hence AB and CD are perpendicular to the same straight line ;^hence they are parallel (Prop. XVIII.), PROPOSITION XXIII. THEOREM. \. Two parallels are every where equally distant. Two parallels AB, CD, being ç hgiven, if through two points Eand F, assumed at pleasure, thestraight lines EG, FH, be drawnperpendicular to AB,these straight ^lines will at the same time beperpendicular to CD (Prop. XX. Cor. 1.) : and we are now toshow that they will be equal to each other. If GF be drawn, the angles GFE, FGH, considered in refer-ence to the parallels AB, CD, will be alternate angles, andtherefore equal to each other (Prop. XX. Cor. 2.). Also, thestraight lines EG, FH, being perpendicular to the same straightline AB, are parallel (Prop. XVIH.) ; and the angles EGF,GFH, considered in reference to the parallels EG, FH, will bealternate angles, and therefore equal. Hence the two trian-gles EFG, FGH, have a common side, and two adjacent anglesin each equal ; hence these triangles arc equal (Prop. VI.) ;therefore, the side EG, which measures the distance of theparallels AB and CD at the point E, is equal to the side
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