The theorem for determining the perpendicularity of a line and a plane: a straight line is perpendicular to two intersecting straight lines in the plane.
The property of plane perpendicularity: If two planes are perpendicular, then the straight line perpendicular to the intersection line on one plane must be perpendicular to the other plane.
The property of line-plane perpendicularity: if one of the two parallel lines is perpendicular to the plane, the other one is also perpendicular to the plane.
Related extensions:
If two straight lines in space are parallel to a third straight line, then the two straight lines are parallel. (This corollary means that the transitivity of parallel lines is not only true in plane geometry, but also in space geometry.)
There is and is only one straight line passing through a point in space (whether on a known plane or not) perpendicular to the plane. Let's discuss how to make this unique straight line.
Choose one of the two faces and draw a straight line perpendicular to the intersection of the two faces. Because they are in the same plane, they can definitely be made. Then, because the lines are perpendicular, the intersecting line is also in the other plane, and the line made is outside the other plane, so the line plane is vertical.
The determination theorem that a straight line is perpendicular to a plane (line-plane perpendicularity theorem): If a straight line is perpendicular to two intersecting straight lines in a plane, then the straight line is perpendicular to the plane.
Given that m∥n and m⊥α, verify n⊥α. Proof: Suppose m∩α=M, n∩α=N. Then take another P and Q on m and n respectively.
∵m∥n
∴Suppose m and n determine the plane β, and α∩β=MN
Through N, AB⊥MN is drawn in α, Connect to PN.
∵PM⊥α, ABα
∴PM⊥AB
∵PMβ, MNβ
∴AB⊥β
∵QNβ
∴QN⊥AB~~~①
Also ∵PM⊥α, MNα
∴PM⊥MN
< p>∵PM∥QN∴QN⊥MN~~~②
∵MN∩AB=N, MNα, ABα
∴QN⊥α< /p>