To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle.
在给定直线上，按给定角作一个平行四边形，使它等于给定三角形。
在给定直线 AB、给定角 D 上贴作平行四边形 BHKL 使其面积等于给定三角形 C。BEFG 为辅助平行四边形（与三角形 C 等积），HK、LM 是把面积“贴上” AB 的延长结构。
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Let AB be the given straight line, C the given triangle and D the given rectilineal angle; thus it is required to apply to the given straight line AB, in an angle equal to the angle D, a parallelogram equal to the given triangle C. Let the parallelogram BEFG be constructed equal to the triangle C, in the angle EBG which is equal to D [I. 42]; let it be placed so that BE is in a straight line with AB; letFG be drawn through to H, and let AH be drawn through A parallel to either BG or EF. [I. 31] Let HB be joined.
先用 euclid-elements/book1-prop-042 作一个等于给定三角形、且有给定角的平行四边形。
Then, since the straight line HF falls upon the parallels AH, EF, the angles AHF, HFE are equal to two right angles. [I. 29] Therefore the angles BHG, GFE are less than two right angles; and straight lines produced indefinitely from angles less than two right angles meet; [Post. 5] therefore HB, FE, when produced, will meet. Let them be produced and meet at K; through the point K let KL be drawn parallel to either EA or FH, [I. 31] and let HA, GB be produced to the points L, M.
把这个平行四边形沿给定直线的一端贴放。
Then HLKF is a parallelogram, HK is its diameter, and AG, ME are parallelograms. and LB, BF the so-called complements, about HK; therefore LB is equal to BF. [I. 43] But BF is equal to the triangle C; therefore LB is also equal to C.
通过给定直线作平行线，使新图形保持同高和给定角。
[C.N. 1] And, since the angle GBE is equal to the angle ABM, [I. 15] while the angle GBE is equal to D, the angle ABM is also equal to the angle D. Therefore the parallelogram LB equal to the given triangle C has been applied to the given straight line AB, in the angle ABM which is equal to D.
由平行四边形补形相等（euclid-elements/book1-prop-043），得到贴在给定直线上的所需平行四边形。