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?

Note by Arron Kau
5 years, 2 months ago

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