Perpendicularity

Semirrecta AB is perpendicular to the CD straight, because the two angles that form are 90 degrees (in orange and blue respectively).

In geometry, the condition of perpendicularity (from the Latin per-pendiculum «plumb line») is when a straight line cuts another forming a right angle, which measures 90°. Perpendicularity is a fundamental property studied in geometry and trigonometry, for example in right triangles, which have 2 "perpendicular" segments.

The notion of perpendicularity is generalized to that of orthogonality.

Relationships

The relationship of perpendicularity can be given between:

• Recipes: two coplanaries are perpendicular when they cut into the plane in four equal regions. Each of which is a straight angle, at the intersection point of two perpendicular straights is called each one's foot on the other.
  • Semirects: two semi-recipes are perpendicular, when they form straight angles having or not the same point of origin.
• Planes: two planes are perpendicular when they make up four dieros angles of 90o.
  • Semiplanes: two semi-planes are perpendicular when they form dieros angles of 90°; generally, sharing the same line of origin.

In addition, there may be a relationship of perpendicularity between the 4 previous elements, taken two by two.

If two lines intersect to form congruent adjacent angles, they are perpendicular. By analogy, if two planes intersect to form congruent adjacent dihedral angles, they are perpendicular. The sides of a dihedral angle and their opposite half planes determine two perpendicular planes.

Perpendicular lines in the plane

For all perpendicular lines of the plane the following is true.

Notation

Given the set R of lines in the plane, we say that two lines a, b of R are perpendicular and we will notice it:

${displaystyle a,bin R:quad abot b}$

The notation being correct:

${displaystyle abot bquad {mbox{o}}quad bot (a,b)quad {mbox{o bien}}quad (a,b)in bot }$

If two lines are not perpendicular we will notice it:

${displaystyle lnot (abot b)quad {mbox{o}}quad not bot (a,b)quad {mbox{o bien}quad (a,b)notin bot }$

Uniqueness Postulate

In a plane, through a point belonging to or outside a line passes one and only one perpendicular line.

Construction of the perpendicular to a line through a given point

Construction of the perpendicular (blue) to the AB line through point P.

To construct a perpendicular to line AB through point P using straightedge and compass, proceed as follows:

• Step 1 (red): a circle with center in P is drawn to create the A' and B' points in line AB, which are equidistant to P.
• Step 2 (green): two circles are drawn focused on A' and B', passing both by P. Sea Q the other intersection point of these two circles.
• Step 3 (blue): P and Q are joined to obtain the PQ perpendicular straight.

To prove that PQ is perpendicular to AB, we use the LLL congruence test for triangles QPA' and QPB' to show that the angles OPA' and OPB' They are equal. Then the LAL criterion is used for the OPA' and OPB' to show that the angles POA and POB are equal.

Properties

The lines a, b of the plane P, fulfill the following properties:

• Reflexive relationship: all straight a of the plane is not perpendicular to itself:

${displaystyle forall ain P:quad lnot (abot a)}$

• Symmetrical ratio: if a straight line a is perpendicular to another b, the straight b is perpendicular to the a:

${displaystyle forall a,bin P:quad abot bquad longrightarrow quad bbot a}$

Regarding parallel lines

The lines a and b are parallel, as seen by the squares, and are cut by the line perpendicular c.

As seen in the figure, if two lines (a and b) are perpendicular to a third line (c), all the angles formed in the third line are right angles. Therefore, in Euclidean Geometry, any pair of lines that are perpendicular to a third line are parallel to each other, due to Euclid's fifth postulate. Conversely, if a line is perpendicular to a second line, it is also perpendicular to any line parallel to the second line.

In the figure, all orange angles are congruent to each other and all green angles are congruent to each other, because vertically opposite angles are congruent and alternate interior angles formed by a cross section of parallel lines are congruent. Therefore, if the lines a and b are parallel, any one of the following conclusions leads to all the others:

• One of the angles of the diagram is a straight angle.
• One of the orange angles is congruent with one of the green angles.
• The line c is perpendicular to the line a.
• The line c is perpendicular to the line b.