G.522 segment right angles

XI.18

If a segment be at right angles to any plane, all the planes through it is at right angles to the same plane.

segment_right_angles()

prove all the planes through / A B / are also at right angles to the plane of reference

assert Let any segment / A B / be at right angles to the plane of reference

construct Let the plane < D E > be drawn through / A B /

construct Let / C E / be the common section of the plane < D E > and the plane of reference, let a point F be taken at random on / C E /

construct perpendicular_at: From F let / F G / be drawn in the plane < D E > at right angles to / C E /.

assert / A B / is at right angles to the plane of reference,

∵ segment_right_angles_defassert / A B / is also at right angles to all the segments which meet it and are in the plane of reference; so that it is also at right angles to / C E /; therefore < A B F ) is right

assert < G F B ) is also right

∵ corresponding_angles_parallelassert / A B / is parallel to / F G /

assert / A B / is at right angles to the plane of reference

∵ segments_parallel_rightassert / F G / is also at right angles to the plane of reference

∵ plane_right_angles_defassert Now a plane is at right angles to a plane, when the segments drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane

assert / F G /, drawn in one of the planes < D E > at right angles to / C E /, the common section of the planes, was proved to be at right angles to the plane of reference

assert the plane < D E > is at right angles to the plane of reference

∴ Similarly also it can be proved that all the planes through / A B / are at right angles to the plane of reference

Dependency Graph

Required for

G.577, G.578