Video Transcript
In the following figure, the line
between 𝐴 and 𝐵 is parallel to the line between 𝐶 and 𝐷, while the line between
𝐸 and 𝐹 cuts the line between 𝐴 and 𝐵 and the line between 𝐶 and 𝐷 at 𝑋 and
𝑌, respectively. Draw straight line 𝑀𝑁, where the
line between 𝑀 and 𝑁 is parallel to the line between 𝐸 and 𝐹 and cuts the line
between 𝐴 and 𝐵 and the line between 𝐶 and 𝐷 at 𝑂 and 𝑃, respectively, on the
right side of the line between 𝐸 and 𝐹. Find the measure of the angle
𝑂𝑃𝑌.
In this question, we’re given a lot
of information about a given figure. We can start by adding the extra
information onto the figure. First, we are told that the lines
between 𝐴 and 𝐵 and 𝐶 and 𝐷 are parallel. This makes the line between 𝐸 and
𝐹 a transversal of this pair of parallel lines.
Next, we’re told that we need to
construct a line between two points, 𝑀 and 𝑁, on the figure, where this line is
parallel to the line between 𝐸 and 𝐹. And it intersects the line between
𝐴 and 𝐵 at 𝑂 and the line between 𝐶 and 𝐷 at 𝑃 and is on the right of the line
between 𝐸 and 𝐹.
To construct this line, we first
need to choose a point of intersection between the line and one of the lines 𝐴𝐵
and 𝐶𝐷. Let’s choose the point 𝑂 as
shown. We need to make sure that this
point is on the line between 𝐴 and 𝐵 and that this point is to the right of the
line between 𝐸 and 𝐹. We can then recall that we can
construct a line parallel to another line through a point by duplicating the angle
at this point. So we will duplicate angle 𝐴𝑋𝐸
at the point 𝑂.
To do this, we first need to trace
a circle centered at 𝑋 that intersects the two sides of the angle as shown. We will call these points of
intersection 𝐴 prime and 𝐸 prime. Next, we trace a congruent circle
centered at 𝑂. And we will call the point of
intersection between the circle and the line that is to the left of 𝑂 𝑋 prime as
shown. We now trace a circle of radius 𝐴
prime 𝐸 prime centered at 𝑋 prime and call the point of intersection between the
circle shown 𝑀.
We can now note that triangle 𝐴
prime 𝑋𝐸 prime is congruent to triangle 𝑋 prime 𝑂𝑀 by the side-side-side
criterion. Hence, angle 𝐴 prime 𝑋𝐸 prime is
congruent to angle 𝑋 prime 𝑂𝑀. We can add to the diagram that
these angles both have measure 80 degrees. We can also call the point of
intersection between line 𝑀𝑂 and line 𝐶𝐷 𝑃 as shown. And we can also add a point onto
the line called 𝑁. We can also note that the line
between 𝐴 and 𝐵 is a transversal of the lines between 𝑋 and 𝑌 and 𝑂 and 𝑃 with
congruent corresponding angles. So they must be parallel.
We want to find the measure of
angle 𝑂𝑃𝑌. And we can mark this angle on the
diagram as shown. We can note that the line between
𝑂 and 𝑃 is a transversal of parallel lines. So the corresponding angles must be
congruent. Hence, we can conclude that the
measure of angle 𝑂𝑃𝑌 must be equal to 80 degrees.
In the previous example, we can
actually note many more useful properties of this type of construction. First, we can note that
quadrilateral 𝑂𝑃𝑌𝑋 has opposite sides parallel. So it is a parallelogram. We also know that diagonally
opposite angles in a parallelogram are congruent. So we can note that the measure of
angle 𝑌𝑋𝑂 must also be 80 degrees. Similarly, we can note that a
straight angle has measure 180 degrees. We can use this to find that the
measure of angle 𝑋𝑂𝑃 is 100 degrees and that the diagonally opposite angle 𝑋𝑌𝑃
must have the same measure. We can use this process to prove
these properties will hold true for any parallelogram.
