Parallel Lines

Two lines are parallel lines if they do not intersect at any point or if they coincide.

The symbols // or || are used to indicate two parallel lines.

$$ r // s $$ $$ r \parallel s $$

The region between the two lines is called a "strip" or "band."

The minimum distance from any point on one line to the other parallel line is known as the "height of the strip" or the distance between the parallel lines.

This height represents the perpendicular distance between the two lines and remains constant at every point along each line.

An Alternative Definition

Two lines (not parallel to the y-axis) are parallel if they have the same slope and vice versa.

Practically, two lines are parallel if they have the same inclination with respect to the x-axis.

Lines parallel to the y-axis are exceptions because their slope is undefined.

Inverse Theorem. If two lines have the same slope (m=m'), then they are parallel.

A Practical Example and Proof

Consider two parallel lines r:y=2x-2 and r':y=2x-6

Let x₁ and x₁' be the points where the lines intersect the x-axis.

Identify two more points on the x-axis by adding a constant k to x₁ and x₁'. For instance, let k=1.

$$ x_2=x_1+k $$

$$ x_2'=x_1'+k $$

Thus, the two segments x₁x₂ and x₁'x₂' are of the same length.

Draw a perpendicular line from x₁+k and x₁'+k intersecting their respective lines r and r'.

This creates two right triangles, ABC and A'B'C'.

These triangles have the right angles (90°) congruent α=α'.

According to the parallel lines theorem, the angles β=β' are also congruent because they are corresponding angles since parallel lines r and r' are cut by the same transversal (the x-axis).

According to the second criterion of congruence, the two triangles ABC≅A'B'C' are congruent because they have one congruent side AB≅A'B' by construction and the adjacent congruent angles α≅α' and β≅β'.

Note. Since these are right triangles, they both have a 90° angle (α=α'). They both have an angle (β=β') determined by the same transversal intersecting the x-axis. Therefore, the third angle must also be congruent (γ=γ') because the sum of the angles in a right triangle is 180°.

The two triangles are congruent ABC≅A'B'C', meaning all their sides are the same length.

Thus, the two legs of the right triangle are congruent.

$$ \overline{AB} \cong \overline{A'B'} $$

$$ \overline{BC} \cong \overline{B'C'} $$

Knowing that AB=x₂-x₁ and A'B'=x'₂-x'₁ (where x₂=x₁+k and x'₂=x'₁+k) and that BC=y₂-y₁ and A'B'=y'₂-y'₁

$$ x_2 - x_1 = x'_2-x'_1 $$

$$ y_2 - y_1 = y'_2-y'_1 $$

The slope is determined by the ratio of the two legs of the right triangle:

$$ m = \frac{y_2-y_1}{x_2-x_1} $$

$$ m' = \frac{y'_2-y'_1}{x'_2-x'_1} $$

Since the sides are equal, the slope is the same.

$$ m = m' $$

Proof of the Inverse Theorem. The reasoning can also be done in reverse. Starting from the hypothesis that two lines with the same slope m=m' can demonstrate that the two lines are parallel.

If two lines have the same slope $$ m=m' $$ since m=BC/AB and m'=B'C'/A'B', the following equality holds $$ \frac{\overline{BC}}{\overline{AB}} = \frac{\overline{B'C'}}{\overline{A'B'}} $$ Where AB and A'B' are segments on the x-axis, while BC and B'C' are segments on the y-axis. If the segments on the x-axis AB≅A'B' are congruent, I find that the segments on the y-axis BC≅B'C' are also congruent.

Therefore, the right triangles ABC≅A'B'C' are congruent according to the first criterion of triangle congruence, because they have two congruent sides AB≅A'B' and BC≅B'C' and the angle between them α≅α' (90°) congruent. Consequently, the triangles ABC≅A'B'C' have all the same angles. In particular, I am interested in knowing that the angles β≅β' are congruent because they measure the inclination of the lines r and r' concerning the x-axis, as this allows me to deduce that the two lines have the same inclination regarding the x-axis, i.e., they are parallel lines r // r'.

The Condition for Parallelism of Lines

Two lines $ r $ and $ s $ with the following equations in implicit form $$ r: \ ax+by+c=0 $$ $$ s: \ a'x+b'y+c'=0 $$ are parallel $ r // s $ if the coefficients of the equations satisfy the following condition $$ ab' - a'b = 0 $$ which can also be written in this equivalent form $$ \frac{a}{a'} = \frac{b}{b'} $$

This condition is known as the parallelism condition for lines.

Example

Consider an example of two parallel lines $ r // s $ with the following equations in implicit form:

$$ r: \ 2x+3y+1=0 $$

$$ s: \ 4x+6y-8=0 $$

Let's verify if the two lines satisfy the parallelism condition.

$$ ab' - a'b = 0 $$

In this case, a=2, b=3, a'=4, and b'=6.

$$ 2 \cdot 6 - 4 \cdot 3 = 0 $$

$$ 12 - 12 = 0 $$

The parallelism condition is satisfied, so the two lines are parallel $ r // s $.

Proof

Consider two parallel lines $ r // s $ with the following equations in implicit form:

$$ r: \ ax+by+c=0 $$

$$ s: \ a'x+b'y+c'=0 $$

The slopes of the two lines are:

$$ m = - \frac{a}{b} $$

$$ m' = - \frac{a'}{b'} $$

Since the lines are parallel by the initial hypothesis, the two slopes are equal:

$$ m = m' $$

$$ - \frac{a}{b} = - \frac{a'}{b'} $$

With a few simple algebraic steps, we get the parallelism condition between the lines:

$$ \frac{a}{b} = \frac{a'}{b'} $$

$$ ab' = a'b $$

And this concludes the proof.

How to Determine if Two Lines are Distinctly Parallel or Coincident

Two lines are distinctly parallel if their equations satisfy the parallelism condition but do not satisfy the coincidence condition. $$ \frac{a}{a'} = \frac{b}{b'} \ne \frac{c}{c'} $$ Conversely, if they satisfy both conditions, they are coincident lines. $$ \frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} $$

Both distinctly parallel lines and coincident lines satisfy the parallelism condition:

$$ \frac{a}{a'} = \frac{b}{b'} $$

Where a,a',b,b' are the coefficients of the variables x and y in the two lines:

$$ r: ax+by+c=0 $$

$$ r': a'x+b'y+c'=0 $$

In other words, coincident lines are a special case of parallel lines because they have the same slope.

To determine if two parallel lines are distinct or coincident, we must also analyze the coincidence condition by examining the ratio of their constant terms $ \frac{c}{c'} $ .

• Coincident
  If the ratio of the constant terms is proportional to that of the coefficients, then the lines are coincident, meaning they have all points in common: $$ \frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} $$
• Distinctly Parallel
  If the ratio of the constant terms is different from that of the coefficients, then the lines are distinctly parallel, meaning they do not intersect at any point: $$ \frac{a}{a'} = \frac{b}{b'} \ne \frac{c}{c'} $$

Example

Consider two lines with the following improper form equations:

$$ r:2x+3y+6=0 $$

$$ r′:4x+6y+12=0 $$

We need to determine if they are intersecting, parallel, or coincident.

First, let's check if the parallelism condition is satisfied:

$$ \frac{a}{a'} = \frac{b}{b'} $$

In this case, the coefficients of the variables x and y are a=2, b=3 in the first equation, and a'=4, b'=6 in the second equation.

$$ \frac{2}{4} = \frac{3}{6} $$

$$ \frac{1}{2} = \frac{1}{2} $$

The parallelism condition is satisfied, so we know for sure that the two lines are not intersecting.

However, we need to determine if they are distinctly parallel or coincident.

To do this, we analyze the coincidence condition by examining the ratio of the constant terms c=6 and c'=12 of the two equations:

$$ \frac{c}{c'} = \frac{6}{12} = \frac{1}{2} $$

All the ratios are equal to 1/2, so the two lines are coincident:

$$ \frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} = \frac{1}{2} $$

In other words, the two lines have all points in common.

These simple steps allow us to accurately determine the relationship between two lines based on their coefficients without solving a system of equations.

Example of Distinctly Parallel Lines. For completeness, let's consider another example where two lines are distinctly parallel. Take these two lines: $$ r:2x+3y+6=0 $$ $$ r′:4x+6y+3=0 $$ The equations are similar to the previous example. The parallelism condition is satisfied: $$ \frac{a}{a'} = \frac{b}{b'} $$ $$ \frac{2}{4} = \frac{3}{6} $$ $$ \frac{1}{2} = \frac{1}{2} $$ However, in this case, the coincidence condition is not satisfied because the ratio of the constant terms is equal to 2: $$ \frac{c}{c'} = \frac{6}{3} = 2 $$ So the ratio c/c' of the constants is not proportional to the ratio of the variable coefficients: $$ \frac{a}{a'} = \frac{b}{b'} = \frac{1}{2} \ne 2 = \frac{c}{c'} $$ Therefore, the two lines are distinctly parallel.

Observations

Some observations about parallel lines:

• Reflexive Property
  A line r is always parallel to itself. $$ r \parallel r $$
• Symmetric Property
  If a line r is parallel to line s, then line s is parallel to line r. $$ r \parallel s $$
• Transitive Property
  If lines r // s are parallel and lines s // p are parallel, then lines r // p are also parallel: $$ r \parallel s \ , \ s \parallel p \ \Rightarrow r \parallel p $$
• The distance between two parallel lines is constant at every point along each line.
  
  

  Proof. If we consider two parallel lines r // s, since lines perpendicular to one line are also perpendicular to the other, if we take two segments AB and CD perpendicular to the lines r and s, they form a rectangle ABCD. In a rectangle, the opposite sides are congruent, so the segments AB≅CD are congruent. This proves that the distance (height) between two parallel lines is constant at any point.
  

• The Parallel Lines Theorem
  If two lines cut by a third transversal line satisfy one of these conditions:
  • They form congruent alternate (interior or exterior) angles
  • They form congruent corresponding (interior or exterior) angles
  • They form supplementary conjugate angles
  then the two lines are parallel.
• Two lines r and s perpendicular to a third line t are parallel
  Lines r and s form pairs of congruent corresponding angles. Therefore, according to the parallel lines theorem, they are parallel.
  

  Note. Alternatively, we can say that lines r and s form pairs of congruent alternate interior (or exterior) angles or pairs of supplementary conjugate angles. In all cases, the conclusion is the parallelism of the two lines.

• The Parallel Postulate (Euclid's Fifth Postulate)
  Given a point P outside of line r, there is exactly one line that runs parallel to r.
  
• A line t perpendicular to line r is also perpendicular to all lines parallel to r
  Given two parallel lines r and s, if line t is perpendicular to one line, it is also perpendicular to the other.
  

  Proof. Consider two parallel lines r//s and a third line t perpendicular to line s. According to the initial hypothesis, lines t and s form a right angle α=90°.
  
  Lines r and s are parallel lines r//s. According to the parallel lines theorem, lines r and s form two pairs of congruent corresponding angles with the same transversal line (α≅β). Therefore, angle β is also a right angle (β=90°).
  
  Consequently, line t is also perpendicular to line r.

• Two lines a and b parallel to a third line c are also parallel to each other (a//b) $$ a \parallel c \ , \ b \parallel c \ \Rightarrow a \parallel b $$

  Proof. To prove this, we use a contradiction. Assume lines a and b are both parallel to line c but lines a and b are not parallel to each other $$ a \parallel c $$ $$ b \parallel c $$ If lines a and b are not parallel, they intersect at a point P.
  
  However, according to the initial hypothesis, both lines a and b are parallel to line c. This means that two lines (a and b) pass through point P and are parallel to line c, which is impossible because it contradicts the parallel postulate. Therefore, lines a and b cannot intersect. Thus, the initial statement is false. Its negation is true, meaning lines a and b are both parallel to line c and are also parallel to each other (a//b)
  

• The parallel lines of two intersecting lines a and b are also intersecting
  

  Proof. By contradiction, given two intersecting lines a and b, their respective parallel lines a' and b' are parallel to each other a' // b' $$ a \parallel a' $$ $$ b \parallel b' $$ $$ a' \parallel b' $$ If lines a' and b' are parallel to each other (a' // b'), according to the transitive property, lines a // b would also be parallel $$ a' \parallel b' \ , \ b' \parallel b \Rightarrow a \parallel b $$ However, this contradicts the initial hypothesis that lines a and b are intersecting. Thus, the statement is false. Its contrary is true, meaning given two intersecting lines a and b, their respective parallel lines a' and b' are also intersecting.

• Theorem of Parallel Planes Intersected by a Third Plane
  When a plane \( \pi \) intersects two parallel planes \( \pi_1 \) and \( \pi_2 \), the intersections form two parallel lines.

  This happens because when a plane \( \pi \) cuts through two parallel planes \( \pi_1 \) and \( \pi_2 \), the intersections create two lines \( r_1 \) and \( r_2 \). Since \( \pi_1 \) and \( \pi_2 \) are equidistant and \( \pi \) intersects them at the same angle, the lines \( r_1 \) and \( r_2 \) are guaranteed to be parallel.

And so on.