Video Transcript
In this video, we will learn how to
calculate angle measures that are created by intersecting lines in a circle. To do that, we’ll review some key
terms we need to know when working with circles, consider three different circle
theorems, and then use that information to solve some examples.
If we have a circle, a chord is a
line segment where both endpoints lie on the circle. A secant is a line that intersects
the circle at two distinct points. The diameter is a line segment,
where both endpoints lie on the circle and the midpoint is the center of the
circle. The diameter is the longest chord
of any circle. The radius is a line segment
connecting the center of the circle to a point on the circle. A tangent is a line that intersects
the circle at only one point, and the tangent makes a right angle with the radius at
that point. Now that we’ve looked at lines and
line segments in a circle, let’s consider angles and arcs.
If you have two radii, the angle
created between the two of them is called a central angle. The part of the circumference
created by this central angle is known as an arc. An arc of the circle is a portion
of the circumference. When we talk about the arc measure,
we’re saying the degree measure of an arc, and it’s equal to the measure of the
central angle that intercepts the arc. For example, if our central angle,
pictured here, is 120 degrees, then we would say that the arc measure is 120
degrees. We can also say that the arc
highlighted in yellow measures 120 degrees. When we do that, we’re saying that
the arc is 120 degrees out of the full 360 degrees of the whole circle. And while we’ll be primarily
focusing on degrees in this video, all of this information is equally true if you’re
operating in radians.
Let’s now consider some of the
theorems, starting with the angles of intersecting chords theorem. Inside the circle, the line segment
𝐵𝐷 or chord 𝐵𝐷 intersects chord 𝐴𝐶 at this point. This intersection creates four
different angles, labeled here one, two, three, and four. Since in this circle, the two
chords 𝐴𝐶 and 𝐵𝐷 intersect inside the circle, the measure of angle one is equal
to one-half times the measure of arc 𝐴𝐵 plus the measure of arc 𝐶𝐷. And likewise, the measure of angle
two, one-half times the measure of arc 𝐵𝐶 plus the measure of arc 𝐴𝐷. Something else that we can say
since vertical angles are congruent is that the measure of angle one is equal to the
measure of angle three and the measure of angle two is equal to the measure of angle
four.
We’ve just looked at angles that
are created when chords intersect. Now, let’s consider what angles are
created when secants intersect. When two secants intersect outside
of the circle, in this case, the intersection is here, they create this angle. In addition to that, it creates two
intercepted arcs. Here, we have the first one in
green and the second one in blue. If we call the angle created by the
two secants angle one, and we’ll call the intercepted arc one 𝑥 and the intercepted
arc two 𝑦, then the measure of angle one will be equal to the measure of the arc 𝑥
minus the measure of the arc 𝑦 divided by two.
If we write this out, we’re saying
that if two lines intersect outside the circle, then the measure of an angle formed
by the two lines is one-half the positive difference of the measures of the
intercepted arcs. In this case, we’re dealing with
two secants that are intersecting outside the circle. But this is also true when two
tangents intersect outside the circle. Let’s see what two tangents
intersecting would look like.
Now, we have two tangents
intersecting. They form this angle that we can
call angle one, which creates intercepted arc one and intercepted arc two. If we let the larger arc be 𝑥 and
the smaller one be 𝑦, we would find the measure of angle one by subtracting the
measure of arc 𝑥 minus the measure of arc 𝑦 and then dividing by two. Using this information, let’s try
and solve for some missing values.
Find the value of 𝑥.
In the given figure, line segment
𝐶𝐸 and line segment 𝐴𝐸 intersect at point 𝐸. The angle created at their
intersection is the one labeled 𝑥. And as these lines have intersected
the circle, they have created two intercepted arcs. If we sketch the center of the
circle, we can show arc 𝐷𝐵 that measures 71 degrees and arc 𝐶𝐴 that measures 144
degrees.
Since the line segment 𝐶𝐸 and the
line segment 𝐴𝐸 can both be described as secants of the circle, we can use the
angles of intersecting secants theorem. Which tells us the angle created by
the intersection of two secants, in this case, the measure of angle 𝐶𝐸𝐴, will be
equal to one-half the difference of the two intercepted arcs. And that means we can say that 𝑥
is equal to 144 minus 71 divided by two. 𝑥 is equal to 36.5 degrees.
Here’s another example. This time, instead of the
intersection of two secants outside the circle, we have the intersection of a secant
and a tangent outside the circle.
Given that, in the shown figure, 𝑦
equals 𝑥 minus two and 𝑧 equals two 𝑥 plus two, determine the value of 𝑥.
First, let’s start with what we
know. Line segment 𝐷𝐴 and line segment
𝐵𝐴 intersect outside the circle at point 𝐴. Because that is true, we can say
the measure of the angle formed by the two lines is one-half the positive difference
of the measures of the intercepted arc. In this case, we already know that
the measure of the angle formed by these two lines is 50 degrees, but that 50
degrees is equal to 𝑧 degrees minus 𝑦 degrees over two. Then, what we can do is substitute
in two 𝑥 plus two in for 𝑧 and 𝑥 minus two in for 𝑦.
Using this equation, we will be
able to find the value of 𝑥. If we distribute the negative to
the 𝑥 and the negative two, we will have 50 equals two 𝑥 plus two minus 𝑥 plus
two divided by two. If we combine like terms, two 𝑥
minus 𝑥 equals positive 𝑥 and two plus two equals four. And so, we can say that 50 equals
𝑥 plus four over two. From there, we can get the two out
of the denominator by multiplying both sides by two. We’ll have 100 equals 𝑥 plus
four. And if 100 equals 𝑥 plus four and
we subtract four from both sides, we see that 𝑥 equals 96.
We know that 𝑦 was equal to 𝑥
minus two. And so, 𝑦 would be 94 degrees. 𝑧 was equal to two 𝑥 plus
two. If we multiply 96 by two and then
add two, we get 194. 𝑧 was then equal to 194
degrees. If we wanted to check, we can plug
these values back in for 𝑧 and 𝑦 in the original equation we wrote, which says 50
degrees equals 𝑧 degrees minus 𝑦 degrees over two. 194 minus 94 divided by two. 194 minus 94 is 100 and 100 divided
by two is 50. And so, we can say that, in the
given figure, 𝑥 must be equal to 96.
Here’s another example.
Determine the measure of arc
𝐶𝐵.
First, let’s see what we know. We know that the line segment 𝐶𝐴
and the line segment 𝐸𝐴 intersect outside the circle at point 𝐴. Both line segment 𝐶𝐴 and line
segment 𝐸𝐴 are secants of the circle. The figure has also told us that
the measure of arc 𝐶𝐵 is equal to the measure of arc 𝐸𝐷. We can also say that the sum of all
arc measures in the circle must be 360 degrees. We’re interested in finding the
measure of arc 𝐶𝐵. But in order to do that, we’ll need
to find the measure of arc 𝐵𝐷.
Based on what we said initially
about the intersection outside the circle, we can say that the angle made by two
intersecting lines outside the circle is half the positive difference of the
intercepted arcs, which is the intersecting secant angles theorem. The angle created by these
intersecting lines in our case is 34 degrees. 34 degrees is equal to the measure
of the intercepted arc 𝐶𝐸, which is 151 degrees, minus the measure of the
intercepted arc 𝐵𝐷 and then divided it by two.
We need to solve for the measure of
the arc 𝐵𝐷. So, we multiply both sides of the
equation by two, which gives us 68 equals 151 minus the measure of arc 𝐵𝐷. So, we subtract 151 from both
sides, and we get negative 83 degrees equals the negative measure of arc 𝐵𝐷. We multiply both sides of the
equation by negative one and flip the sides, and we get the measure of arc 𝐵𝐷
equals positive 83 degrees. But this is not our final
answer. We’re still trying to find the
measure of arc 𝐶𝐵.
But remember, we know that all of
these arcs must sum to 360 degrees. And we know that arc 𝐶𝐵 and arc
𝐸𝐷 must be equal. We could say that they’re equal to
𝑥 degrees. If we do that, we could then create
the equation 151 plus 𝑥 plus 𝑥 plus 83 equals 360. If we combine like terms, 151 plus
83 equals 234. 𝑥 plus 𝑥 equals two 𝑥. Therefore, 234 plus two 𝑥 equals
360 degrees. We subtract 234 from both sides of
the equation, which tells us two 𝑥 equals 126. Divide both sides by two, and we
see that 𝑥 equals 63. This tells us that both arc 𝐶𝐵
and arc 𝐸𝐷 is equal to a measure of 63 degrees. We were primarily interested in the
measure of arc 𝐶𝐵, and we found that that measure is 63 degrees.
In our next example, we have two
chords that intersect inside a circle.
In the given figure, find the
measure of arc 𝐴𝐶 plus the measure of arc 𝐵𝐷.
Let’s see what we know. By looking at the figure, we can
see that line segment 𝐵𝐴 and line segment 𝐶𝐷 are chords that intersect inside a
circle, which means we can think about the angles of intersecting chords
theorem. Which tells us that the measure of
angle one is equal to one-half the measure of arc 𝑃𝑆 plus the measure of arc
𝑄𝑅. In our figure, we’re interested in
the measure of arc 𝐴𝐶 plus the measure of arc 𝐵𝐷.
Based on what we know about
intersecting chords in a circle, we can say that 112 degrees is equal to one-half
times the measure of arc 𝐴𝐶 plus the measure of arc 𝐵𝐷. And if we multiply both sides of
this equation by two, we would have 224 degrees is equal to the measure of arc 𝐴𝐶
plus the measure of arc 𝐵𝐷. And that’s what we’re looking for,
the measure of arc 𝐴𝐶 plus the measure of arc 𝐵𝐷, which is 224 degrees.
In this example, we haven’t been
given an image. And so, we’ll have to sketch one on
our own.
The point 𝐴 is outside a circle
with center 𝑀. The line between 𝐴 and 𝐶
intersects the circle at 𝐵 and 𝐶, and the line between 𝐴 and 𝐸 meets the circle
at points 𝐷 and 𝐸. Given that the measure of angle
𝐶𝑀𝐸 equals 130 degrees and the measure of angle 𝐵𝑀𝐷 is equal to 56 degrees,
find the measure of angle 𝐴.
We’ve been given a description of
our circle and lines, but we haven’t been given an image. The best place to start here is
with sketching. We know we have a circle and that
point 𝐴 is outside the circle. We also know that there is a line
between 𝐴 and 𝐶; there’s a line with endpoints 𝐴 and 𝐶. And this line intersects the circle
at 𝐵 and 𝐶. If we draw a line from 𝐴 to the
circle, we know that the endpoints of this line were 𝐴 and 𝐶, and the other
intersection along the circle was point 𝐵.
Similarly, we have a line between
𝐴 and 𝐸 that meets the circle at 𝐷 and 𝐸. We’ll draw another line from point
𝐴. The endpoint is 𝐸 and its other
intersection point is 𝐷. The circle has a center 𝑀. We’ve been told the measure of
angle 𝐶𝑀𝐸, which would be this angle, is 130 degrees. And we’ve been told the measure of
angle 𝐵𝑀𝐷, which would be this angle, and that measures 56 degrees. And we want to know the measure of
angle 𝐴.
Now, of course, when we look at our
sketch, we know that these angles are a little bit off. But this sketch gives us enough
information to figure out how we’re going to try and solve for the measure of angle
𝐴. Because we have two lines
intersecting outside of a circle, then the angle created by the two lines
intersecting outside the circle is half the positive difference between the
intercepted arcs.
And that means the measure of angle
𝐴 is the angle created outside the circle. It’s going to be one-half the
measure of arc 𝐶𝐸 minus the measure of arc 𝐷𝐵. Arc 𝐶𝐸 measures 130 degrees; arc
𝐵𝐷 measures 56 degrees. 130 minus 56 equal 74, and half of
74 is 37. And so, a circle under these
conditions will have the angle measure 𝐴 equal to 37 degrees.
And now, we can review the key
points. The intersecting secant angle
theorem tells us that the angle made by two secants intersecting outside a circle is
half the positive difference between the intercepted arc measures. This is true for two secants, two
tangents, and when one secant and one tangent intersect on the outside of a
circle. And then, we have our intersecting
chords theorem, which tells us that the measure of angle one equals one-half the
measure of arc 𝐴𝐵 plus the measure of arc 𝐶𝐷. And by extension, the measure of
angle two equals one-half times the measure of arc 𝐵𝐶 plus the measure of arc
𝐴𝐷.
And because we know the definitions
of vertical angles, we know that when chords intersect, the measure of angle one is
equal to the measure of angle three, and the measure of angle two is equal to the
measure of angle four.
