### Video Transcript

The given figure shows a pair of
parallel lines and two transversals, one of which crosses at right angles. Write an expression for π in terms
of π. Using this expression for π, find
a fully simplified expression for π in terms of π.

Letβs see what we have. We have two parallel lines. We could call them line one and
line two, and then we have two transversals, which we could call line three and line
four. The first thing we wanna do is
write an expression for π in terms of π. π and π occur at the intersection
of line one and line three. They are adjacent angles on the
same side of line one. This makes them supplementary
angles. They must add up to 180
degrees. If we know that π degrees plus π
degrees must equal 180 degrees and we want π in terms of π, this means we want to
try to get π by itself and have π on the other side of the equation. To do that, we can subtract π
degrees from both sides of the equation, which means π degrees will be equal to 180
degrees minus π degrees, and therefore π will be equal to 180 minus π. This completes part one, our first
expression.

Now, using this expression for π,
we need to find a fully simplified expression for π in terms of π, which means we
first need to look at the relationship between π degrees and π degrees. When it comes to π and π, they
are in between the two parallel lines and on the same side of the transversal. They are consecutive interior
angles, sometimes called cointerior angles. And when two parallel lines are
crossed by a transversal, the consecutive interior angles are supplementary. They sum to 180. And that means π degrees plus π
degrees must equal 180 degrees.

For this expression, we want to
express π in terms of π, and that means weβll need to get π by itself. First, we subtract π from both
sides, which gives us π degrees equals 180 degrees minus π degrees or π equals
180 minus π. But remember, we want to substitute
our expression weβve already found for π. We know that π equals 180 minus
π. Remember though, weβre subtracting
all of π, so we need to keep that in parentheses and then distribute that
subtraction, which means subtract 180. But subtracting negative π would
be adding π, and that means π equals 180 minus 180 plus π. 180 minus 180 equals zero, and zero
plus π equals π. A fully simplified expression for
π in terms of π is that π is equal to π.

In fact, if we take a closer look
at π and π, we see that the relationship between these two angles are alternate
interior angles, and we know that alternate interior angles will be congruent. So π in terms of π would be π
equals 180 minus π, and π in terms of π would be π equals π.