Posts tagged 'parallel'
G.52 parallel lines
Parallel lines are lines in the same plane which, extended indefinitely in both directions, do not meet.
G.53 equal alternate interior angles imply parallel lines
If a transversal cutting two lines makes the alternate interior angles equal, the lines are parallel.
G.54 equal corresponding angles imply parallel lines
If a transversal cutting two lines makes the exterior angle equal to the opposite interior angle, or interior angles on the same side supplementary, the lines are parallel.
G.56 transversal angles of parallel lines
A transversal cutting parallel lines makes alternate interior angles equal, exterior and opposite interior angles equal, and same-side interior angles supplementary.
G.57 lines parallel to the same line are parallel
Lines parallel to the same line are parallel to one another.
G.58 construct a line parallel to a given line
Construct through a given point a line parallel to a given line.
G.60 segments joining equal and parallel segments
Segments joining the endpoints of equal and parallel segments in the same directions are themselves equal and parallel.
G.64 parallelograms on the same base in the same parallels
Parallelograms on the same base and in the same parallels are equal in area.
G.65 parallelograms on equal bases in the same parallels
Parallelograms on equal bases and in the same parallels are equal in area.
G.66 triangles on the same base in the same parallels
Triangles on the same base and in the same parallels are equal in area.
G.67 triangles on equal bases in the same parallels
Triangles on equal bases and in the same parallels are equal in area.
G.68 equal triangles on the same base are in the same parallels
Equal triangles on the same base and on the same side lie in the same parallels.
G.69 equal triangles on equal bases are in the same parallels
Equal triangles on equal bases and on the same side lie in the same parallels.
G.70 parallelogram is double a triangle on the same base
If a parallelogram shares the same base with a triangle and lies in the same parallels, the parallelogram is double the triangle.
G.211 segment drawn parallel
If a segment be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of the triangle be cut proportionally, the line joining the points of section is parallel to the remaining side of the triangle.
G.245 triangles sides proportional
If two triangles having two sides proportional to two sides be placed together at one angle so that their corresponding sides are also parallel, the remaining sides of the triangles is in a segment.
G.508 segments right angles
If two segments be at right angles to the same plane, the segments is parallel.
G.509 segments parallel points
If two segments be parallel and points be taken at random on each of them, the segment joining the points is in the same plane with the parallel lines.
G.510 segments parallel right
If two segments be parallel, and one of them be at right angles to any plane, the remaining one is at right angles to the same plane.
G.511 straight lines parallel
Straight lines which are parallel to the same segment and are not in the same plane with it are also parallel to one another.
G.512 segments meeting parallel
If two segments meeting one another be parallel to two segments meeting one another not in the same plane, they will contain equal angles.
G.516 parallel planes
Parallel planes are those which do not meet.
G.517 planes segment right
Planes to which the same segment is at right angles is parallel.
G.518 segments meeting parallel
If two segments meeting one another be parallel to two segments meeting one another, not being in the same plane, the planes through them are parallel.
G.519 parallel planes cut
If two parallel planes be cut by any plane, their common sections are parallel.
G.520 segments cut parallel
If two segments be cut by parallel planes, they is cut in the same ratios.
G.530 solid contained parallel
If a solid be contained by parallel planes, the opposite planes in it are equal and parallelogrammic.
G.531 parallelepipedal solid cut
If a parallelepipedal solid be cut by a plane which is parallel to the opposite planes, then, as the base is to the base, so will the solid be to the solid.
G.537 prism
A:term:**prism** <prism> is a:term:solid figure <G.529> contained by:term:planes <G.7> two of which, namely those which are opposite, are equal, similar and:term:parallel <G.52>, while the rest are parallelograms.
G.571 cylinder cut plane
If a cylinder be cut by a plane which is parallel to its opposite planes, then, as the cylinder is to the cylinder, so will the axis be to the axis.