Perpendicular and parallel lines

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   M : y = cx + d

as long as neither is vertical. Then a and c are the slopes of the two lines. The lines L and M are perpendicular if and only if the product of their slopes is -1, or if ac = − 1.

The angle times the height of another angle equals the sum of one angle. The perpendicular force is equivalent to the base and also the height of the vertex/reflex angle(s). Perpendicular lines have negative reciprocal slopes. So perpendicular means a 90 degree angle at at least two sides of the symbol. It has to have a line sticking straight up above a horisontal line.

To construct the perpendicular to the line AB through the point P using compass and straightedge, proceed as follows (see Figure 2).

   * Step 1 (red): construct a circle with center at P to create points A' and B' on the line AB, which are equidistant from P.
   * Step 2 (green): construct circles centered at A' and B', both having a radius greater than A'P. Let Q be the other point of intersection of these two circles.
   * Step 3 (blue): connect P and Q to construct the desired perpendicular PQ.

To prove that the PQ is perpendicular to AB, use the SSS congruence theorem for triangles QPA' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the SAS congruence theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal.

As shown in Figure 3, if two lines (a and b) are both perpendicular to a third line (c), all of the angles formed on the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.

In Figure 3, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because vertical angles are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines a and b are parallel, any of the following conclusions leads to all of the others:

   * One of the angles in the diagram is a right angle.
   * One of the orange-shaded angles is congruent to one of the green-shaded angles.As shown in Figure 3, if two lines (a and b) are both perpendicular to a third line (c), all of the angles formed on the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.

If lines a and b are parallel, any of the following conclusions leads to all of the others:

   * One of the angles in the diagram is a right angle.
   * One of the orange-shaded angles is congruent to one of the green-shaded angles.
   * Line 'c' is perpendicular to line 'a'.
   * Line 'c' is perpendicular to line 'b'.
   * Line 'c' is perpendicular to line 'a'.
   * Line 'c' is perpendicular to line 'b'.

Algebra

In algebra, for any linear equation y=mx + b, the perpendiculars will all have a slope of (-1/m), the opposite reciprocal of the original slope. It is helpful to memorize the slogan "to find the slope of the perpendicular line, flip the fraction and change the sign." Recall that any whole number a is itself over one, and can be written as (a/1)

To find the perpendicular of a given line which also passes through a particular point (x, y), solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b.

Calculus

First find the derivative of the function. This will be the slope (m) of any curve at a particular point (x, y). Then, as above, solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b.

In the Unicode character set, the perpendicular sign has the codepoint U+27C2 and is part of the Miscellaneous Mathematical Symbols-A range: ⟂.


Parallel Lines[edit]

Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. Just remember: Parallel lines always the same distance apart and never touch.

Parallel lines also point in the same direction.

When parallel lines get crossed by another line (which is called a Transversal), you can see that many angles are the same.

These angles can be made into pairs of angles which have special names.