Video Transcript
In this video, we will learn how to
apply the angle relationships between a pair of parallel lines and a transversal to
establish and use other relationships between parallel lines and transversals. There are many uses for the angle
relationships between parallel lines and transversals. The most common use is finding the
measures of other angles using results such as the alternate interior angles being
congruent. However, there are other uses, such
as proving relationships between transversals of lines.
To see this in action, we first
recall that a transversal of a pair of parallel lines will have congruent
corresponding angles. This is a useful result, since we
can use this to determine the angles between two lines using the angle between one
line and a parallel line. In particular, if we have a line
𝐴𝐵 that is perpendicular to another line 𝐸𝐹, then we know that any line parallel
to 𝐴𝐵 must have a corresponding angle with a measure of 90 degrees. This gives us the following
result. If a line is perpendicular to line
𝐴𝐵, then it is perpendicular to any line parallel to line 𝐴𝐵. This is not the only property we
can show using properties of transversals.
We recall that a transversal of a
pair of parallel lines will have congruent corresponding angles. We can also recall that if the
corresponding angles in a transversal of two lines are congruent, then the lines are
parallel. Therefore, if a transversal of two
lines is perpendicular to the two lines, then the lines must be parallel, since they
have congruent corresponding angles. This gives us the following
property. If two lines are perpendicular to
the same line, then they must be parallel. Let’s now see an example of
applying these properties to determine relationships between lines.
If line 𝐴𝐵 is parallel to
line 𝐶𝐷 and line 𝐸𝐹 is perpendicular to line 𝐶𝐷, which of the following is
correct? (A) Line segment 𝐸𝐺 bisects
line segment 𝐴𝐵. (B) Line 𝐴𝐵 is parallel to
line 𝐸𝐺. (C) Line 𝐴𝐵 is perpendicular
to line 𝐶𝐷. (D) Line 𝐴𝐵 is perpendicular
to line 𝐸𝐺. (E) Line segment 𝐸𝐹 bisects
line segment 𝐶𝐷.
Let’s begin by adding the fact
that line 𝐸𝐹 being perpendicular to line 𝐶𝐷 means the lines are at right
angles and the fact that lines 𝐴𝐵 and 𝐶𝐷 are parallel to our diagram. We can then use corresponding
angles to note that line 𝐸𝐹 is perpendicular to line 𝐴𝐵. This shows that line 𝐴𝐵 is
perpendicular to line 𝐸𝐺, which is answer (D). However, this is a particular
case of the fact that if a line is perpendicular to line 𝐿, then it is
perpendicular to any line parallel to 𝐿.
It is worth noting that
although the diagram looks like the transversal bisects the parallel lines, this
does not need to be, since we can translate the transversal and still have the
image being perpendicular to the parallel lines. We can see that line 𝐿 is
perpendicular to the parallel lines 𝐴𝐵 and 𝐶𝐷 and that it is not a
perpendicular bisector of either line. Thus, we cannot conclude that
line segment 𝐸𝐺 bisects line segment 𝐴𝐵 or that it bisects line segment
𝐶𝐷. So answers (A) and (E) are
incorrect. Hence, line 𝐴𝐵 is
perpendicular to line 𝐸𝐺, and the correct answer is option (D).
Before we move on to our next
example, there is another useful property we can show. Let’s consider a pair of distinct
parallel lines, 𝐴𝐵 and 𝐶𝐷, and another pair of distinct parallel lines, 𝐶𝐷 and
𝐸𝐹. It appears as though all three of
the lines are parallel. We can prove this is the case by
sketching a transversal perpendicular to line 𝐸𝐹. Using corresponding angles, we can
show that all three lines are perpendicular to the transversal. Then, we can recall that if two
lines are perpendicular to the same line, then they must be parallel to each
other. Hence, all three lines are
parallel.
We have proven the following
result. If two distinct lines are parallel
to a third distinct line, then all three lines are parallel. Let’s now see an example of using
this property to determine the relationship between given lines.
Fill in the blank. If line 𝐴𝐵 is parallel to
line 𝐶𝐷 and line 𝐴𝐵 is parallel to line 𝐸𝐹, then line 𝐶𝐷 is what to line
𝐸𝐹.
We first recall that if two
lines are parallel to the same line, then they must be parallel. Since both lines 𝐶𝐷 and 𝐸𝐹
are parallel to the same line, 𝐴𝐵, we must have that the two lines are
parallel. Hence, the answer is that line
𝐶𝐷 is parallel to line 𝐸𝐹, denoted as shown.
There is one final property that we
want to show about the transversals of parallel lines. And we will introduce this property
with an example.
Consider the following diagram,
where 𝐿 one, 𝐿 two, and 𝐿 three are all parallel and the lengths of the line
segments between the parallel lines are as shown. In the diagram, it appears that if
the lengths of the line segments of a transversal between three parallel lines are
equal, then the lengths of the line segments of any transversal between three
parallel lines will be equal. We can prove this by using
congruent triangles.
We can start with a perpendicular
transversal that is split into two sections of equal length by the parallel lines as
shown. We can note that the measure of
angle 𝑀𝑁𝐿 is equal to the measure of angle 𝑂𝑁𝑃, as they are vertically
opposite angles. We now see that triangle 𝑂𝑁𝑃 and
triangle 𝑀𝑁𝐿 are congruent by the ASA criterion. In particular, this means that 𝐿𝑁
is equal to 𝑁𝑃. Hence, the other transversal is
also split into two sections of equal length. We can always add in a transversal
perpendicular to these lines. So the proof of this result in
general is very similar.
We have shown the following
property. If a set of parallel lines divide a
transversal into segments of equal length, then they divide any other transversal
into segments of equal length. Let’s now see some examples of
using this property to find the length of a transversal between parallel lines.
If lines 𝐴𝐵, 𝐶𝐷, and 𝐸𝐹
are all parallel and 𝐶𝐸 equals two centimeters, find 𝐴𝐸.
We can see in the diagram that
we are given three parallel lines and two transversals. We can also see that 𝐵𝐷 is
equal to 𝐷𝐹. So, in this case, the parallel
lines divide one of the transversals into line segments of equal length. We can recall that if a set of
parallel lines divide a transversal into segments of equal length, then they
divide any other transversal into segments of equal length. Therefore, it will divide the
other transversal into sections of equal length. Thus, 𝐴𝐶 is equal to 𝐶𝐸,
which is equal to two centimeters. We note that 𝐴𝐸 is equal to
𝐴𝐶 plus 𝐶𝐸. So 𝐴𝐸 is equal to two
centimeters plus two centimeters, which equals four centimeters.
In our final example, we will apply
multiple properties of the transversals of parallel lines to a triangle with a side
bisected by parallel lines.
Consider triangle 𝐴𝐵𝐶 and
lines 𝐴𝑀 and 𝐸𝐷, which are parallel to line 𝐶𝐵. Find the length of the line
segment 𝐴𝐵. Find the measure of angle
𝐴𝐵𝐶.
We are given three parallel
lines and two transversals of these lines. We can then recall that if a
set of parallel lines divide a transversal into segments of equal length, then
they divide any other transversal into segments of equal length. Since 𝐴𝐸 is equal to 𝐸𝐶,
the segments of the other transversal must be equal in length. So 𝐴𝐷 is equal to 𝐷𝐵, which
equals five millimeters. Since 𝐴𝐵 is equal to 𝐴𝐷
plus 𝐷𝐵, then 𝐴𝐵 equals five millimeters plus five millimeters, which equals
10 millimeters. 𝐴𝐵 is equal to 10
millimeters.
It appears in the diagram that
triangle 𝐴𝐵𝐶 is a right triangle. However, we need to justify why
this is the case. We can do this by recalling
that if a line is perpendicular to line 𝐿, then it is perpendicular to any line
parallel to 𝐿. Since line 𝐸𝐷 is
perpendicular to line 𝐴𝐶 and line 𝐸𝐷 is parallel to line 𝐵𝐶, we must have
that lines 𝐵𝐶 and 𝐴𝐶 are perpendicular. This means the angle at 𝐶 has
a measure of 90 degrees, so 𝐴𝐵𝐶 is a right triangle.
The sum of the measures of the
interior angles in a triangle is 180 degrees. So 180 degrees equals 35
degrees plus 90 degrees plus the measure of angle 𝐴𝐵𝐶. Rearranging the equation, we
have the measure of angle 𝐴𝐵𝐶 equals 180 degrees minus 35 degrees minus 90
degrees, which equals 55 degrees. The answers to the two parts of
this question are 10 millimeters and 55 degrees.
Let’s finish by recapping some of
the most important points from this video. If a line is perpendicular to line
𝐴𝐵, then it is perpendicular to any line parallel to line 𝐴𝐵. If two lines are perpendicular to
the same line, then they must be parallel. If two distinct lines are parallel
to a third distinct line, then all three lines are parallel. If a set of parallel lines divide a
transversal into segments of equal length, then they divide any other transversal
into segments of equal length.
