### Video Transcript

In this video, weβll learn how to
use the parallel chords in a circle and parallel tangents and chords to deduce the
equal measures of the arcs between them and find missing lengths or angles. Letβs begin by recapping some of
the key terminology for circles. Consider this circle, whose center
is at point π. A chord is a line segment whose
endpoints lie on the circumference of the circle. In the diagram, line segment π΄π΅,
which is represented with a bar above the letters π΄π΅, is a chord to the
circle. Then, a tangent to a circle is a
line that intersects the circle exactly once. In the diagram here, the line πΆπ·,
which is represented with a two-headed arrow above πΆπ·, is a tangent to the circle
at point π.

When the lines are added to a
circle, the points where they meet the circle partition its circumference into a
number of arcs. For instance, there are two arcs
between points π΄ and π΅. The short arc is known as the minor
arc π΄π΅. This is the arc whose measure is
less than 180 degrees. And we use the arc notation above
the letters π΄π΅ to represent this. Traveling from π΄ to π΅ in the
opposite direction, then the arc has a measure greater than 180 degrees, and itβs
called a major arc. One way that we can differentiate
between the minor arc and the major arc is to include point π when we represent the
major arc. We could write this as arc
π΄ππ΅.

So with these definitions in mind,
we can define a theorem that links parallel chords and arcs in a circle. This theorem tells us that the
measure of the arcs between parallel chords of a circle are equal. In this diagram, since π΄π΅ is
parallel to line segment π·πΆ, the measure of arc π΄πΆ β thatβs here β is equal to
the measure of arc π΅π·. What this also means of course is
that these two arcs are congruent. And thatβs really useful for
solving problems.

Now, whilst itβs outside the scope
of this video to prove this theorem, with some knowledge of the angles and parallel
lines cut by a transversal and inscribed angles in a circle, it can be proved in
just a few steps. So letβs apply this theorem
alongside some other properties of chords to find the measure of an arc.

In the given figure, if the measure
of arc π΅π· equals 65 degrees, find the measure of arc πΆπ·.

In the diagram, we notice that
weβve been given a pair of parallel chords. That is, line segment π΄π΅ is
parallel to line segment πΆπ·. And we recall that arcs formed by a
pair of parallel chords are congruent. So arc π΄πΆ and arc π΅π· are
congruent, which means the measures of these arcs must be equal. And so we see that the measure of
arc π΄πΆ is 65 degrees.

Next, we see that line segment π΄π΅
in fact passes through the center of the circle. It must therefore be the diameter
of the circle. And so it splits this circle
exactly in half. And so we can say that the measure
of both arcs π΄π΅ are 180 degrees. Now, of course, weβre interested in
the portion of the circle which passes through points πΆ and π·. So weβve called that the measure of
arc π΄πΆπ·π΅. The question wants us to find the
measure of arc πΆπ·. And we now know that the measure of
all the individual arcs between π΄ and π΅ is 180 degrees. So we can say that the sum of the
measure of arc π΄πΆ, the measure of arc πΆπ·, and the measure of arc π·π΅ is
180. But remember, we said that arcs
π΄πΆ and π·π΅ are congruent and their measures are 65 degrees. So our equation becomes 65 degrees
plus the measure of arc πΆπ· plus another 65 degrees equals 180. And then we simplify that left-hand
side.

We can now solve this equation for
the measure of arc πΆπ· by subtracting 130 from both sides. So itβs 180 minus 130, which is of
course 50. So the measure of arc πΆπ· is 50
degrees.

Now that weβve demonstrated this
theorem, there is another property that holds for parallel chords of equal
lengths. That is, if two chords are parallel
and equal in length, then the arcs between the endpoints of those chords will be
equal in measure. In the diagram, the chords π΄π΅ and
πΆπ· are parallel and of equal length. So, somewhat intuitively, the
measure of arc π΄π΅ has to be equal to the measure of arc πΆπ·. In our next example, weβll
demonstrate how to apply this property.

In the diagram, the measure of arc
π΄π΅ is equal to 62 degrees, the measure of arc π΅πΆ equals 110 degrees, and the
measure of arc π΄π· equals 126 degrees. What can we conclude about the line
segments π΄π· and π΅πΆ? (A) They are parallel. (B) They are neither parallel nor
perpendicular. (C) They are perpendicular. (D) Theyβre the same length. (E) They are parallel and of the
same length.

Letβs begin by adding the measure
of each of our arcs to the diagram. Weβre told the measure of arc π΄π΅
equals 62 degrees, the measure of arc π΅πΆ equals 110, and the measure of arc π΄π·
equals 126. Since the measure of each arc is
the angle that arc makes at the center of the circle, it follows that the sum of all
arc measures that make up that circle is 360 degrees. And this is really useful because
it will allow us to calculate the measure of arc πΆπ·.

We say that the sum of the arc
measures is 360 degrees. And then we can replace the various
arc measures with their values. So the measure of arc π΄π΅ is 62,
the measure of arc π΅πΆ is 110, and so on. This left-hand side simplifies to
298 degrees plus the measure of arc πΆπ·. And then we can find that measure
of arc πΆπ· by subtracting 298 from both sides. Itβs 360 minus 298, which is of
course 62 degrees.

So why is this useful? Well, we know that the measure of
arcs between parallel chords of a circle are equal, and the opposite is also
true. That is, if the measure of two arcs
between two distinct chords is equal, those chords must in fact be parallel. We in fact have that the measure of
arc π΄π΅ is equal to the measure of arc πΆπ·. And so that must mean that line
segments π΄π· and π΅πΆ are in fact parallel. Weβre therefore able to disregard
options (B), (C), and (D). And so we need to choose between
option (A) and option (E), where option (A) is that they are parallel and option (E)
is that theyβre not only parallel, but theyβre of the same length.

Well, if the chords are parallel
and equal in length, then the arcs between the endpoints of the chords will be equal
in measure. But we can see that the measure of
arc π΅πΆ is not equal to the measure of arc π΄π·. They are in fact 110 and 126
degrees, respectively. So π΅πΆ and π΄π· cannot be of equal
length. And so the answer is (A). Theyβre parallel.

In this example, we showed that the
reverse statement to our earlier theorem holds. If the measure of the two arcs
between two distinct chords is equal, then the chords themselves must be
parallel. Weβre now going to extend our idea
of parallel chords to include a parallel chord and a tangent.

The next theorem states that the
measure of the arcs between a parallel chord and tangent of a circle are equal. In this diagram, the line segment
or chord π΄π΅ is parallel to the tangent at πΆ. And so we can say that the measure
of arc π΄πΆ must be equal to the measure of arc π΅πΆ. With this theorem stated, letβs
demonstrate its application.

π is a circle, where line segment
π΄π΅ is a chord and line πΆπ· is a tangent. If π΄π΅ is parallel to πΆπ· and the
measure of arc π΄π΅ is 72 degrees, find the measure of arc π΅πΆ.

Since π΄π΅ is parallel to πΆπ·,
where π΄π΅ is a chord and πΆπ· is a tangent, thereβs a theorem we can use. Weβre going to use the theorem that
says that the measure of the arcs between a parallel chord and tangent of a circle
are equal. So we can say that the measure of
arc π΄πΆ must be equal to the measure of arc π΅πΆ. In fact, the question tells us that
the measure of arc π΄π΅ is 72 degrees. And we can use the fact that the
sum of all the measures of all arcs which make up the circle is 360 degrees. This means that the measure of arc
π΄πΆ plus the measure of arc π΄π΅, which is 72 degrees, plus the measure of arc π΅πΆ
is 360. Then, we subtract 72 degrees from
both sides. And we find that the measure of arc
π΄πΆ plus the measure of arc π΅πΆ is 288 degrees.

But earlier, we stated that the
measure of arc π΄πΆ is equal to the measure of arc π΅πΆ. So we can say that two times the
measure of arc π΅πΆ is 288 degrees. And then we could divide both sides
of this equation by two. So the measure of arc π΅πΆ is 288
divided by two, which is in fact 144 degrees. The measure of arc π΅πΆ is 144
degrees.

In the examples weβve seen so far,
weβve applied the theorems of parallel chords and tangents in a circle to find
missing values given information about their chords and tangents. Now, itβs useful to remember that
these properties can be applied alongside geometric properties of polygons to help
us find missing values. Letβs demonstrate this.

In the following figure, a
rectangle π΄π΅πΆπ· is inscribed in a circle, where the measure of arc π΄π΅ equals 71
degrees. Find the measure of arc π΄π·.

Weβre going to use the fact that
π΄π΅πΆπ· is a rectangle. This means that the line segment or
chord π΄π΅ is parallel to chord π·πΆ. Similarly, line segment π·π΄ must
be parallel to line segment π΅πΆ. This means we can use the theorem
that tells us that the measure of arcs between parallel chords of a circle are
equal. Since line segments π΄π΅ and π·πΆ
are parallel, the measure of arc π΄π· must be equal to the measure of arc π΅πΆ. Similarly, the measure of arc π΄π΅
must be equal to the measure of arc π·πΆ. But weβre actually told thatβs 71
degrees.

Now, since the sum of all the arc
measures that make up the circle is 360 degrees, we can form and solve an
equation. We know that the measure of arc
π΄π΅ and the measure of arc π·πΆ is 71. So our equation is the measure of
arc π΄π· plus the measure of arc π΅πΆ plus 71 plus 71 equals 360. Since π΄π· and π΅πΆ are congruent
arcs, we can further simplify this. Two times the measure of arc π΄π·
plus 142 degrees equals 360 degrees. Then, we subtract 142 degrees from
both sides, and our final stage is to divide through by two. So the measure of arc π΄π· is 218
divided by two, which is equal to 109 or 109 degrees. The measure of arc π΄π· is 109
degrees.

In our final example, weβll
demonstrate how to apply the theorems of parallel chords and tangents to allow us to
solve problems involving algebraic expressions for arc measures.

In the following figure, π΄π΅ and
πΈπΉ are two equal chords. π΅πΆ and πΉπΈ are two parallel
chords. If the measure of arc π΄πΆ is 120
degrees, find the measure of arc πΆπΈ.

Letβs begin by using the fact that
these two line segments π΄π΅ and πΈπΉ are two equal chords. Since theyβre equal in length, we
can deduce that the measure of their arcs must also be equal. So the measure of arc π΄π΅ must be
equal to the measure of arc πΈπΉ. In fact, weβre told that this is
equal to π₯ degrees. Then, we use the information about
π΅πΆ and πΉπΈ; theyβre parallel chords. This means that the measures of the
arcs between those two chords is equal. That is, the measure of arc πΆπΈ
must be equal to the measure of arc π΅πΉ. And this time weβre also told that
that is equal to π₯ plus 30 degrees.

Using this information alongside
the measure of arc π΄πΆ, we know that the sum of all the arc measures is 360
degrees. So we can form and solve an
equation. The sum of the arcs is π₯ plus π₯
plus 30 plus π₯ plus π₯ plus 30 plus 120. And that must be equal to 360. And so that left-hand side
simplifies to four π₯ plus 180. So four π₯ plus 180 degrees equals
360. We can therefore say that four π₯
must be equal to 180. And we can then solve for π₯ by
dividing through by four. So π₯ degrees equals 45
degrees. We want to find the measure of arc
πΆπΈ, and we said that that was equal to π₯ plus 30. So the measure of arc πΆπΈ is 45
plus 30, which is equal to 75 degrees. The measure of arc πΆπΈ then is 75
degrees.

Letβs now recap the key points from
this lesson. In this video, we learned that the
measure of the arcs between parallel chords of a circle are equal. We also learned that if two chords
are parallel and equal in length, then the arcs between the endpoints of each chord
will also be equal in measure. And finally, we learned that the
measure of the arcs between a parallel chord and a tangent of a circle are
equal.