The method of determining the perpendicularity of the plane, the determination of the perpendicular

The line and surface are perpendicular, and according to theorems and inferences, it can be obtained

The decision theorem that a straight line is perpendicular to a plane (line-plane perpendicular theorem): a straight line is perpendicular to both intersecting lines in a plane, then the line is perpendicular to the plane.

Corollary 1: If one of the two parallel lines is perpendicular to a plane, then the other line is also perpendicular to this plane.

Corollary 2: If two straight lines are perpendicular to the same plane, then the two straight lines are parallel.

You start with the theorem and find the corresponding conditions to determine it.

Prove that two planes are perpendicular, usually by proving that the line is perpendicular and the line and surface are perpendicular, in the argument about the perpendicular problem, it is necessary to pay attention to the mutual transformation between the three, and if necessary, auxiliary lines can be added, such as: when the surface is known to be perpendicular, the property theorem is generally used to make the perpendicular line of the intersection in a plane, so that it is transformed into the perpendicular of the line and the perpendicular of the line, so that it is necessary to be proficient in the transformation conditions and common methods between the three The perpendicular and perpendicular of the line and the perpendicular of the surface are finally summarized as the perpendicular of the line. The inverse theorem of the Pythagorean theorem and the properties of isosceles triangles are commonly used to prove that two straight lines are perpendicular to coplanarity; Two straight lines that are not coplanar are perpendicular to each other, usually using the line-plane perpendicular or using space vectors

Common conclusions: 1) If two planes are perpendicular to each other, then a straight line perpendicular to the second plane through a point in the first plane is in the first plane, this conclusion can be used as a property theorem, 2) from the conditions of the property theorem: as long as a point in one plane is made a perpendicular line in the other plane, then the perpendicular line must be in this plane, and the position of the point can be on the intersection line or not.

The determination that the plane is perpendicular to the plane is as follows:

1. Prove that the dihedral angle is 90 degrees;

2. Prove that a straight line in the plane is perpendicular to the other plane, then the two planes are perpendicular.

If two planes intersect, the resulting dihedral angle (a figure consisting of two half-planes from a straight line) is a straight dihedral angle (the plane angle is a right angle), and the two planes are said to be perpendicular. If one plane passes through one perpendicular line of the other plane, then the two planes are perpendicular to each other. (Line, Vertical, Vertical).

Proving that two planes are perpendicular is usually achieved by proving that the line is perpendicular and the line and surface are perpendicular, and in the argument on the perpendicular problem, it is necessary to pay attention to the mutual transformation between the three, and if necessary, auxiliary lines can be added, such as: when it is known that the surface is perpendicular, the property theorem is generally used to make the perpendicular line of the intersecting line in a plane, so that it is transformed into the perpendicular line of the line surface, and then converted into the perpendicular of the line and line.

Therefore, it is necessary to be proficient in the transformation conditions and common methods between the three, and the line-plane perpendicularity and the perpendicularity of the surface surface are finally summarized as the perpendicular line line, and the inverse theorem of the Pythagorean theorem and the properties of isosceles triangle are commonly used in the perpendicular of two straight lines in the coplanarity. Two straight lines that are not coplanar are perpendicular to each other, usually using the line-plane perpendicular or using space vectors

Straight Lines and PlanesA positional relationship between straight lines and planes in vertical space. If a line is perpendicular to any two intersecting lines in a plane, the line and the plane are said to be perpendicular to each other. A straight line is called the perpendicular line of a plane, and a flat surface is called the perpendicular surface of a straight line.

The intersection of the straight line and the plane is called the perpendicular foot.

The plane perpendicular to the plane is determined that if one plane passes through a perpendicular line of the other plane, then the two planes are perpendicular to each other.

If the projection of any point in one plane in another plane is on the intersection of the two planes, then it is perpendicular. If one of the n parallel planes is perpendicular to one plane, then the rest are perpendicular to this plane.

Relationship between planes:Straight lines on different planes do not have the conditions to determine the plane. Opposite-plane straight lines neither intersect nor parallel.

It should not be mistaken for two straight lines in different planes as straight lines of different planes. As shown in the figure, although there are a and b, that is, a and b are in two different planes, but because a b o, a and b are not heterogeneous straight lines.

The decision theorems and properties of plane perpendicularity are as follows:

When one plane passes through the perpendicular line of the other, the two planes are perpendicular to each other.

Corollary 1: If the perpendicular line of one plane is parallel to the other, then the two planes are perpendicular to each other.

Corollary 2: If the perpendiculars of two planes are perpendicular to each other, then the two planes are perpendicular to each other. (It can be understood that the planes perpendicular to the normal vector are perpendicular to each other).

Surface-perpendicular properties theorem 1:

If the two planes are perpendicular to each other, then a straight line perpendicular to their intersection in one plane is perpendicular to the other.

Theorem 2: If two planes are perpendicular to each other, then a straight line perpendicular to the second plane is made through a point in the first plane.

Theorem 3: If both two intersecting planes are perpendicular to the third plane, then their intersecting lines are perpendicular to the third plane.

Corollary: The intersection of three planes perpendicular to each other is perpendicular to each other.

Theorem 4: If two planes are perpendicular to each other, then the perpendiculars of one plane are parallel to the other. (Inverse theorem of decision theorem inference 1).

Corollary: If two planes are perpendicular to each other, then the two perpendicular lines that are respectively perpendicular to each other are also perpendicular to each other. (Inverse theorem of decision theorem inference 2).

The method of determining the perpendicularity of the plane is as follows: if the dihedral angle formed by the two planes is 90°, then the two planes are perpendicular; If one plane passes through one perpendicular line of the other plane, then the two planes are perpendicular to each other. If the projection of any point in one plane in another plane is on the intersection of the two planes, then it is perpendicular. The method of determining that the plane is perpendicular to the plane is as follows:

If two planes are formed. If the dihedral angle is 90°, then the two planes are perpendicular; If one plane passes through one perpendicular line of the other plane, then the two planes are perpendicular to each other. If the projection of any point in one plane in another plane is on the intersection of the two planes, then it is perpendicular.