Perpendicular straight lines

Perpendicular straight lines . Two lines in the plane are perpendicular if they form a right angle between them (strictly speaking, four right angles are formed). In a coordinate system, the product of the slopes of both lines is -1.

Summary

[ disguise ]

• 1 Definition of Line
• 2 Definitions of Perpendicular Lines
  • 1 Properties of Perpendicular Lines
• 3 Theorems
• 4 See also
• 5 Sources

Definition of Straight

A Line is an infinite succession of points, all located in the same direction, while this succession is characterized by being continuous and indefinite, therefore, a line has neither beginning nor end; Along with the Plane and the Point , the Line is one of the fundamental geometric Entities.

• A line is an infinite succession of points, located in the same direction.
  • A line has only one dimension: Length.
  • Lines are named by two of their points or by a lowercase letter.
  • Two points determine a line.

Definitions of Perpendicular Lines

• Two lines are perpendicular when when they intersect they form four equal 90º angles.
• Two lines are perpendicular if their direction vectors are perpendicular.
• Given a Point belonging to a line or exterior to it, one and only one perpendicular to said Line passes through it .

Two lines are perpendicular if their direction vectors are perpendicular, that is, the product of the vectors is equal to zero.

If two lines are perpendicular, their slopes are inverse and have changed signs.

The perpendicularity relationship can be given between:

• Lines: two coplanar lines are perpendicular when, when cut, they divide the plane into four equal regions, each of which is a right angle. The point of intersection of two perpendicular lines is called the foot of each of them in the other.
• Semi-straight lines: two semi-straight lines are perpendicular when they form right angles, whether or not they have the same point of origin.
• Planes: two planes are perpendicular when they make up four 90º dihedral angles.
• Halfplanes: two halfplanes are perpendicular when they make dihedral angles of 90°; generally, sharing the same line of origin.

Furthermore, there may be a relationship of perpendicularity between the four previous elements, taken two by two.

If two lines form congruent adjacent angles when they intersect , they are perpendicular. By analogy, if two planes when cut form adjacent congruent dihedral angles, they are perpendicular. The sides of a dihedral angle and its opposite half-planes determine two perpendicular planes).

DEFINITION.- A right triangle is a triangle whose angle is right. The side opposite the right angle is called the hypotenuse and the other two sides are the legs .

DEFINITION.- A line and a plane are perpendicular if they intersect and furthermore, every line in the plane that passes through the point of intersection is perpendicular to the given line.

Properties of Perpendicular Lines

The properties they have are:

• Reflexive: Perpendicularity does not comply with the reflective character.
• Symmetrical: If one line is perpendicular to another, this one is perpendicular to the first.
• Transitive: Perpendicularity does not comply with the transitive character.

Theorems

• Theorem: In a plane, two lines perpendicular to a third are parallel.
• Theorem: In a given plane and through a given point of a given line, one and only one line passes perpendicular to the given line.
• Bisector Theorem: In a given plane, the bisector of a segment is the line perpendicular to the segment at its midpoint.
• Theorem: From a given external point, there is at least one line perpendicular to the given line.

COLORING: No Triangle has two right Angles .

• Theorem – If B and C are equidistant from P and Q then every point between B and C is also equidistant from P and Q.

COROLLARY: A segment AB (with a line above) and the line L are given in the same plane. If two points of L are equidistant from A and B, then the bisector of AB (with dash above).

• Theorem – If a line is perpendicular to two lines that intersect at their point of intersection, then it is perpendicular to the plane that contains the lines.