Euclid's Elements of geometry, books i. ii. iii. iv1862 |
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Página 3
... Parallel straight lines are such as are in the same plane , and which being produced ever so far both ways do not meet . 35. A parallelogram is a four - sided figure , of which the opposite sides are parallel ; and the diagonal is the ...
... Parallel straight lines are such as are in the same plane , and which being produced ever so far both ways do not meet . 35. A parallelogram is a four - sided figure , of which the opposite sides are parallel ; and the diagonal is the ...
Página 27
... parallel . ( References - Prop . I. 16 ; def . 34. ) Hypothesis . Let the straight line EF , which falls upon the two straight lines AB , CD , make the alternate angles AEF , EFD , equal to one another . Sequence . - AB shall be parallel ...
... parallel . ( References - Prop . I. 16 ; def . 34. ) Hypothesis . Let the straight line EF , which falls upon the two straight lines AB , CD , make the alternate angles AEF , EFD , equal to one another . Sequence . - AB shall be parallel ...
Página 28
... parallel . ( def . 34. ) 6. Therefore AB is parallel to CD . Conclusion . Therefore , if a straight line , & c . Q. E. D. PROPOSITION 28. - THEOREM . If a straight line falling upon two other straight lines make the exterior angle equal ...
... parallel . ( def . 34. ) 6. Therefore AB is parallel to CD . Conclusion . Therefore , if a straight line , & c . Q. E. D. PROPOSITION 28. - THEOREM . If a straight line falling upon two other straight lines make the exterior angle equal ...
Página 29
... parallel to CD . ( I. 27. ) Conclusion . - Therefore , if a straight line , & c . Q. E. D. PROPOSITION 29. - THEOREM . If a straight line fall upon two parallel straight lines , it makes the alternate angles equal to one another , and ...
... parallel to CD . ( I. 27. ) Conclusion . - Therefore , if a straight line , & c . Q. E. D. PROPOSITION 29. - THEOREM . If a straight line fall upon two parallel straight lines , it makes the alternate angles equal to one another , and ...
Página 30
... parallel straight lines . ( hyp . ) 8. Therefore the angle AGH is not unequal to the angle GHD ; that is , it is ... parallel to the same straight line are parallel to one another . ( References - Prop . I. 27 , 29 ; ax . 1. ) Hypothesis ...
... parallel straight lines . ( hyp . ) 8. Therefore the angle AGH is not unequal to the angle GHD ; that is , it is ... parallel to the same straight line are parallel to one another . ( References - Prop . I. 27 , 29 ; ax . 1. ) Hypothesis ...
Términos y frases comunes
AB is equal AC and CB adjacent angles angle ABC angle AGH angle BAC angle BCD angle EAB angle EDF angle equal angles CBA base BC BC is equal bisected circle ABC circumference Conclusion Conclusion.-Therefore const Construction.-1 Demonstration.-1 describe the circle diameter double equal angles equal to CD equiangular exterior angle given circle given point given rectilineal angle given straight line Given.-Let ABCD gnomon greater Hypothesis inscribed interior and opposite isosceles triangle less opposite angle parallel to CD parallelogram perpendicular point F produced Q. E. D. PROPOSITION rectangle AB BC rectangle AE rectangle contained rectilineal figure References-Prop remaining angle required to describe right angles segment semicircle Sequence side BC square on AC straight line AC straight line drawn touches the circle triangle ABC triangle DEF twice the rectangle
Pasajes populares
Página 25 - If two triangles have two angles of the one equal to two angles of the other, each to each ; and one side equal to one side, viz.
Página 2 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
Página 99 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.
Página 4 - If a straight line meets two straight lines, so as to " make the two interior angles on the same side of it taken " together less than two right angles...
Página 66 - ... the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle between the perpendicular and the obtuse...
Página 65 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.
Página 32 - F, which is the common vertex of the triangles ; that is, together with four right angles. Therefore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Página 58 - If a straight line be bisected, and produced to any point ; the rectangle contained by the whole line thus produced, and the part of it produced...
Página 88 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...
Página 33 - The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel.