Video Transcript
In this video, we’ll 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. Let’s begin by recapping some of
the key terminology for circles. Consider this circle, whose center
is at point 𝑚. A chord is a line segment whose
endpoints lie on the circumference of the circle. In the diagram, line segment 𝐴𝐵,
which is represented with a bar above the letters 𝐴𝐵, 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 𝐶𝐷,
which is represented with a two-headed arrow above 𝐶𝐷, is a tangent to the circle
at point 𝑃.
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 𝐴 and 𝐵. The short arc is known as the minor
arc 𝐴𝐵. This is the arc whose measure is
less than 180 degrees. And we use the arc notation above
the letters 𝐴𝐵 to represent this. Traveling from 𝐴 to 𝐵 in the
opposite direction, then the arc has a measure greater than 180 degrees, and it’s
called a major arc. One way that we can differentiate
between the minor arc and the major arc is to include point 𝑃 when we represent the
major arc. We could write this as arc
𝐴𝑃𝐵.
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 𝐴𝐵 is
parallel to line segment 𝐷𝐶, the measure of arc 𝐴𝐶 — that’s here — is equal to
the measure of arc 𝐵𝐷. What this also means of course is
that these two arcs are congruent. And that’s really useful for
solving problems.
Now, whilst it’s 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 let’s 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 𝐵𝐷 equals 65 degrees, find the measure of arc 𝐶𝐷.
In the diagram, we notice that
we’ve been given a pair of parallel chords. That is, line segment 𝐴𝐵 is
parallel to line segment 𝐶𝐷. And we recall that arcs formed by a
pair of parallel chords are congruent. So arc 𝐴𝐶 and arc 𝐵𝐷 are
congruent, which means the measures of these arcs must be equal. And so we see that the measure of
arc 𝐴𝐶 is 65 degrees.
Next, we see that line segment 𝐴𝐵
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 𝐴𝐵 are 180 degrees. Now, of course, we’re interested in
the portion of the circle which passes through points 𝐶 and 𝐷. So we’ve called that the measure of
arc 𝐴𝐶𝐷𝐵. The question wants us to find the
measure of arc 𝐶𝐷. And we now know that the measure of
all the individual arcs between 𝐴 and 𝐵 is 180 degrees. So we can say that the sum of the
measure of arc 𝐴𝐶, the measure of arc 𝐶𝐷, and the measure of arc 𝐷𝐵 is
180. But remember, we said that arcs
𝐴𝐶 and 𝐷𝐵 are congruent and their measures are 65 degrees. So our equation becomes 65 degrees
plus the measure of arc 𝐶𝐷 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 𝐶𝐷 by subtracting 130 from both sides. So it’s 180 minus 130, which is of
course 50. So the measure of arc 𝐶𝐷 is 50
degrees.
Now that we’ve 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 𝐴𝐵 and
𝐶𝐷 are parallel and of equal length. So, somewhat intuitively, the
measure of arc 𝐴𝐵 has to be equal to the measure of arc 𝐶𝐷. In our next example, we’ll
demonstrate how to apply this property.
In the diagram, the measure of arc
𝐴𝐵 is equal to 62 degrees, the measure of arc 𝐵𝐶 equals 110 degrees, and the
measure of arc 𝐴𝐷 equals 126 degrees. What can we conclude about the line
segments 𝐴𝐷 and 𝐵𝐶? (A) They are parallel. (B) They are neither parallel nor
perpendicular. (C) They are perpendicular. (D) They’re the same length. (E) They are parallel and of the
same length.
Let’s begin by adding the measure
of each of our arcs to the diagram. We’re told the measure of arc 𝐴𝐵
equals 62 degrees, the measure of arc 𝐵𝐶 equals 110, and the measure of arc 𝐴𝐷
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 𝐶𝐷.
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 𝐴𝐵 is 62,
the measure of arc 𝐵𝐶 is 110, and so on. This left-hand side simplifies to
298 degrees plus the measure of arc 𝐶𝐷. And then we can find that measure
of arc 𝐶𝐷 by subtracting 298 from both sides. It’s 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 𝐴𝐵 is equal to the measure of arc 𝐶𝐷. And so that must mean that line
segments 𝐴𝐷 and 𝐵𝐶 are in fact parallel. We’re 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 they’re not only parallel, but they’re 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 𝐵𝐶 is not equal to the measure of arc 𝐴𝐷. They are in fact 110 and 126
degrees, respectively. So 𝐵𝐶 and 𝐴𝐷 cannot be of equal
length. And so the answer is (A). They’re 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. We’re 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 𝐴𝐵 is parallel to the tangent at 𝐶. And so we can say that the measure
of arc 𝐴𝐶 must be equal to the measure of arc 𝐵𝐶. With this theorem stated, let’s
demonstrate its application.
𝑚 is a circle, where line segment
𝐴𝐵 is a chord and line 𝐶𝐷 is a tangent. If 𝐴𝐵 is parallel to 𝐶𝐷 and the
measure of arc 𝐴𝐵 is 72 degrees, find the measure of arc 𝐵𝐶.
Since 𝐴𝐵 is parallel to 𝐶𝐷,
where 𝐴𝐵 is a chord and 𝐶𝐷 is a tangent, there’s a theorem we can use. We’re 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 𝐴𝐶 must be equal to the measure of arc 𝐵𝐶. In fact, the question tells us that
the measure of arc 𝐴𝐵 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
𝐴𝐶 plus the measure of arc 𝐴𝐵, which is 72 degrees, plus the measure of arc 𝐵𝐶
is 360. Then, we subtract 72 degrees from
both sides. And we find that the measure of arc
𝐴𝐶 plus the measure of arc 𝐵𝐶 is 288 degrees.
But earlier, we stated that the
measure of arc 𝐴𝐶 is equal to the measure of arc 𝐵𝐶. So we can say that two times the
measure of arc 𝐵𝐶 is 288 degrees. And then we could divide both sides
of this equation by two. So the measure of arc 𝐵𝐶 is 288
divided by two, which is in fact 144 degrees. The measure of arc 𝐵𝐶 is 144
degrees.
In the examples we’ve seen so far,
we’ve applied the theorems of parallel chords and tangents in a circle to find
missing values given information about their chords and tangents. Now, it’s useful to remember that
these properties can be applied alongside geometric properties of polygons to help
us find missing values. Let’s demonstrate this.
In the following figure, a
rectangle 𝐴𝐵𝐶𝐷 is inscribed in a circle, where the measure of arc 𝐴𝐵 equals 71
degrees. Find the measure of arc 𝐴𝐷.
We’re going to use the fact that
𝐴𝐵𝐶𝐷 is a rectangle. This means that the line segment or
chord 𝐴𝐵 is parallel to chord 𝐷𝐶. Similarly, line segment 𝐷𝐴 must
be parallel to line segment 𝐵𝐶. 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 𝐴𝐵 and 𝐷𝐶
are parallel, the measure of arc 𝐴𝐷 must be equal to the measure of arc 𝐵𝐶. Similarly, the measure of arc 𝐴𝐵
must be equal to the measure of arc 𝐷𝐶. But we’re actually told that’s 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
𝐴𝐵 and the measure of arc 𝐷𝐶 is 71. So our equation is the measure of
arc 𝐴𝐷 plus the measure of arc 𝐵𝐶 plus 71 plus 71 equals 360. Since 𝐴𝐷 and 𝐵𝐶 are congruent
arcs, we can further simplify this. Two times the measure of arc 𝐴𝐷
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 𝐴𝐷 is 218
divided by two, which is equal to 109 or 109 degrees. The measure of arc 𝐴𝐷 is 109
degrees.
In our final example, we’ll
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, 𝐴𝐵 and
𝐸𝐹 are two equal chords. 𝐵𝐶 and 𝐹𝐸 are two parallel
chords. If the measure of arc 𝐴𝐶 is 120
degrees, find the measure of arc 𝐶𝐸.
Let’s begin by using the fact that
these two line segments 𝐴𝐵 and 𝐸𝐹 are two equal chords. Since they’re equal in length, we
can deduce that the measure of their arcs must also be equal. So the measure of arc 𝐴𝐵 must be
equal to the measure of arc 𝐸𝐹. In fact, we’re told that this is
equal to 𝑥 degrees. Then, we use the information about
𝐵𝐶 and 𝐹𝐸; they’re parallel chords. This means that the measures of the
arcs between those two chords is equal. That is, the measure of arc 𝐶𝐸
must be equal to the measure of arc 𝐵𝐹. And this time we’re also told that
that is equal to 𝑥 plus 30 degrees.
Using this information alongside
the measure of arc 𝐴𝐶, 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 𝑥 plus 𝑥
plus 30 plus 𝑥 plus 𝑥 plus 30 plus 120. And that must be equal to 360. And so that left-hand side
simplifies to four 𝑥 plus 180. So four 𝑥 plus 180 degrees equals
360. We can therefore say that four 𝑥
must be equal to 180. And we can then solve for 𝑥 by
dividing through by four. So 𝑥 degrees equals 45
degrees. We want to find the measure of arc
𝐶𝐸, and we said that that was equal to 𝑥 plus 30. So the measure of arc 𝐶𝐸 is 45
plus 30, which is equal to 75 degrees. The measure of arc 𝐶𝐸 then is 75
degrees.
Let’s 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.
