Parallel Lines

Definition

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

Property A

Given any point A A on line a a, the minimum distance to line b 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 c intersects lines a a and b 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 a and b b are parallel.

Property C

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

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

Property D

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

Proof: By property B, EAB=ACD,EBA=EDC \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 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 A, B are points on line a a, and C,D C, D are points on line b b, then [ABC]=[ABD] [ABC] = [ABD]. (Note: [PQRS] [PQRS] represents the area of figure PQRS PQRS.) The converse is also true.

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

 

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

From property D, we know that triangle EAB,ECD EAB, ECD are similar. Hence, the ratio of their side lengths is the same. Thus EAEB=ECED=ACBD \frac {EA}{EB} = \frac {EC}{ED} = \frac {AC}{BD}, which allows us to calculate that BD=9106=15 BD = \frac {9 \cdot 10}{6} = 15. Remark: Does it matter if point E E is between the two lines, or on the same side of both lines? Draw both versions and compare?

 

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

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

 

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

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

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

Note by Arron Kau
5 years, 8 months ago

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