Parallel Lines

Definition

In 2-dimensional Euclidean geometry, two lines $a$ and $b$ are parallel if they do not intersect. We are familiar with several properties of parallel lines $a$ and $b$:

Property A

Given any point $A$ on line $a$, the minimum distance to line $b$ is a constant. The converse is true; if the minimum distance between two lines is a constant, then they are parallel.

Property B

If line $c$ intersects lines $a$ and $b$, then the corresponding angles of intersection are the same. The converse is true; if the corresponding angles of intersection are the same, then $a$ and $b$ are parallel.

Property C

If line $c$ intersects lines $a$ and $b$, then the alternate interior angles are the same. The converse is true; if the corresponding alternate interior angles are the same, then $a$ and $b$ are parallel.

This is a useful property of parallel lines involving similar triangles.

Property D

If $A, B$ are points on line $a$, and $C, D$ are points on line $b$, let lines $AC, BD$ intersect at $E$. Then, triangles $EAB$ and $ECD$ are similar. The converse is also true. If $EAC$ and $EBD$ are straight lines, and triangles $EAB, ECD$ are similar, then $AB$ is parallel to $CD$.

Proof: By property B, $\angle EAB = \angle ACD, \angle EBA = \angle EDC$, so these 2 triangles have 3 corresponding angles equal. Thus, they are similar. (Students are asked to Test Yourself by proving the converse)

Remark: It does not matter if point $E$ is between the two lines, or on the same side of both lines.

This is a useful property of parallel lines, involving area of a triangle.

Property E

If $A, B$ are points on line $a$, and $C, D$ are points on line $b$, then $[ABC] = [ABD]$. (Note: $[PQRS]$ represents the area of figure $PQRS$.) The converse is also true.

Proof: The triangles have the same base $AB$, and have the same height from property A. Thus, they have the same area. (Students are asked to Test Yourself by proving the converse.)

Technique

In a parallelogram, what is the sum of 2 consecutive angles?

Let $\alpha$ and $\beta$ be 2 consecutive angles in a parallologram. From property B, $\alpha$ will be equal to the angle supplementary to $\beta$, or that $\alpha = 180^\circ - \beta$. Thus, the sum of 2 consecutive angles is $180^\circ$. (Pop quiz: Can you make a similar statement regarding trapeziums?)

$AB$ and $CD$ are 2 parallel lines. $AC$ and $BD$ intersect at $E$. If $EA=6, EB=10$ and $AC=9$, what is the length $BD?$

From property D, we know that triangle $EAB, ECD$ are similar. Hence, the ratio of their side lengths is the same. Thus $\frac {EA}{EB} = \frac {EC}{ED} = \frac {AC}{BD}$, which allows us to calculate that $BD = \frac {9 \cdot 10}{6} = 15$. Remark: Does it matter if point $E$ is between the two lines, or on the same side of both lines? Draw both versions and compare?

$AB$ and $CD$ are 2 parallel lines. $AC$ and $BD$ intersect at $E$. If $[AED] = 11$, what is $[BCE]$?

From property E, $[ABD] = [ABC]$. Then, we may add (or subtract, if $E$ is between the two lines) triangle $EAB$ to get that $[BCE]=[AED]=11$.

Point $X$ is given between 2 parallel lines $a$ and $b$. The base of the perpendicular from $X$ to $a$ is denoted as $A$, the base of the perpendicular from $X$ to $b$ is denoted as $B$. Show that the points $X, A, B$ lie on the same line.

Through $X$, construct line $x$ that is parallel to $a$. Label another point $Y$ on $x$. Then, by Worked Example 1, $180^\circ = \angle AXY + 90^\circ$ and $180^\circ = \angle BXY + 90^\circ$. This gives us $\angle AXY = 90 ^ \circ, \angle BXY = 90^\circ$. Thus, $\angle AXY + \angle BXY = 180^\circ$, which shows that the points $X, A, B$ lie on the same line.

Remark : Is this statement still true if $X$ is on the same side of both lines?