Video Transcript
In this video, we will learn how to
identify the angle of tangency in a circle and find its measure using the measure of
its subtended angle, inscribed angle, or central angle subtended by the same
arc. We will begin by recalling that a
tangent line to a circle is a line that intersects the circle at just one point. We also note that the line segment
from the point of intersection 𝐴 to the center of the circle 𝑀 is a radius of the
circle. And this radius is perpendicular to
the tangent line.
In this video, we want to discuss
angles of tangency. If we consider a tangent to a
circle that meets with a chord of the circle at the point 𝐴 as shown, then the
angle 𝜃 between the chord and the tangent is known as the angle of tangency. Calculating this angle can be done
with the help of several theorems and angle properties. During this video, we will also
need to keep in mind some of the properties of triangles, for example, isosceles and
equilateral triangles. Specifically, because the radii of
a circle all have the same length, two radii can form the sides of an isosceles
triangle as shown. This can be useful as it tells us
that the measures of angles 𝑀𝐴𝐵 and 𝑀𝐵𝐴 are equal.
Additionally, we recall the
inscribed angle theorem or central angle theorem. We recall that two points on a
circle 𝐴 and 𝐵 divide a circle into two arcs, a major arc and a minor arc, noting
that when the arcs are the same length, they divide the circle into two semicircular
arcs. We can then form an inscribed angle
with any point 𝐶 on the major arc as shown. This leads us to the inscribed
angle theorem, which states let 𝐴 and 𝐵 be two points on a circle, 𝑀 be the
center of the circle, and 𝐶 be any point on the major arc. Then, the measure of the central
angle 𝐴𝑀𝐵 is double the measure of the inscribed angle 𝐴𝐶𝐵. Another way of phrasing this is
that the measure of the central angle subtended by two points on a circle is twice
the measure of the inscribed angle subtended by those points.
Having recalled this theorem, let’s
now learn a new theorem that deals with angles of tangency in a circle. The alternate segment theorem
states the following. Let 𝐴 and 𝐵 be two points on a
circle and 𝐶 be the point where a tangent passing through 𝐸, 𝐶, and 𝐷 intersects
the circle. Then, the angles of tangency 𝐴𝐶𝐷
and 𝐵𝐶𝐸 are equal to the angles in the alternate segments 𝐴𝐵𝐶 and 𝐵𝐴𝐶,
respectively.
This theorem can be proved as
follows. We recall that a tangent of the
circle at point 𝐶 forms a right angle with the radius 𝑀𝐶 at the point of
intersection. Labeling one of the angles of
tangency 𝐴𝐶𝐷 as 𝜃, then the measure of angle 𝐴𝐶𝑀 must be 90 degrees minus
𝜃. And since we have an isosceles
triangle, the measure of angle 𝐶𝐴𝑀 must also be equal to 90 degrees minus 𝜃. Since angles in a triangle sum to
180 degrees, the measure of angle 𝐴𝑀𝐶 plus 90 degrees minus 𝜃 plus 90 degrees
minus 𝜃 must equal 180 degrees. The left-hand side of this equation
simplifies as shown. We can then subtract 180 degrees
and add two 𝜃 to both sides such that the measure of angle 𝐴𝑀𝐶 is two 𝜃. Since the measure of the central
angle is two 𝜃, using the inscribed angle theorem, we can conclude that the measure
of angle 𝐴𝐵𝐶 is half the central angle two 𝜃 and is therefore equal to 𝜃. This confirms that angle 𝐴𝐶𝐷 is
equal to angle 𝐴𝐵𝐶.
Let’s now look at an example where
we can use this theorem.
Given that line 𝐵𝐶 is a tangent
to the circle, find the measure of angle 𝐴𝐵𝐶.
We will begin by labeling angle
𝐴𝐵𝐶 as 𝜃. This angle is an angle of tangency,
as 𝜃 lies between a chord and tangent. Since 𝐵𝐶 is a tangent to the
circle and 𝐴, 𝐵, and 𝐷 are three points on the circle, we can use the alternate
segment theorem. This states that an angle of
tangency is equal to the angle in the alternate segment. In this case, the measure of angle
𝐴𝐵𝐶 is equal to the measure of angle 𝐴𝐷𝐵. We are told that this is equal to
78 degrees. And we can therefore conclude that
the measure of angle 𝐴𝐵𝐶 is equal to 78 degrees via the alternate segment
theorem.
Let’s now consider how angles of
tangency relate to central angles. When proving the alternate segment
theorem, we saw that the angle of tangency is half the central angle subtended by
the same arc. This can be more formally stated as
follows. Let 𝐴 be a point on a circle of
center 𝑀 and 𝐵 be the point where a tangent passing through 𝐵 and 𝐶 intersects
the circle. Then, the angle of tangency 𝐴𝐵𝐶
is half of the central angle 𝐴𝑀𝐵. Let’s now look at an example where
we can use this.
Find the measure of angle
𝐵𝐴𝐶.
Let’s begin by marking the angle we
are trying to find on the diagram. The angle we are trying to find is
an angle of tangency, as it lies between a tangent and a chord. We are told that angle 𝐴𝑀𝐵 is a
right angle and is therefore equal to 90 degrees. This is also the central angle
subtended by the same arc 𝐴𝐵 as the angle of tangency. And we recall that the angle of
tangency is equal to a half the central angle. This means that the measure of
angle 𝐵𝐴𝐶 is equal to a half of the measure of angle 𝐴𝑀𝐵. The angle of tangency is therefore
equal to a half of 90 degrees, which is equal to 45 degrees.
An alternative method here would be
to recognize that triangle 𝐴𝑀𝐵 is isosceles. This means that angles 𝐵𝐴𝑀 and
𝐴𝐵𝑀 are equal in measure. And since the angles in a triangle
sum to 180 degrees, they must be equal to 45 degrees. We can then use the fact that a
tangent is perpendicular to the radius at the point of contact, which means that 𝜃
plus 45 degrees equals 90 degrees. And subtracting 45 degrees from
both sides, we have 𝜃 is equal to 45 degrees. This confirms that the measure of
angle 𝐵𝐴𝐶 is 45 degrees.
So far in this video, we have seen
how to calculate an angle of tangency using a central angle and using an inscribed
angle. We can also calculate angles of
tangency using the measure of an arc. We recall that the measure of an
arc is the angle that the arc makes at the center of the circle. For example, in the diagram drawn,
the measure of the major arc of the circle created by the two points 𝐴 and 𝐵 is
equivalent to the measure of the angle between the radii formed by 𝐴 and 𝐵 in the
inside of the circle.
Recalling that the central angle of
a circle is double the angle of tangency, then we can extend this to apply to the
arc of a circle as shown. Since both the measure of the
central angle and arc measure are two 𝜃, they are both twice the angle of
tangency. More formally, if we let 𝐴 be a
point on a circle and 𝐵 be the point where a tangent passing through 𝐵 and 𝐶
intersects the circle, then the angle of tangency 𝐴𝐵𝐶 is half of the measure of
the arc 𝐴𝐵 formed on the same side. It is important to note that this
holds for both the minor and major arcs as shown.
Let’s now look at an example where
we can use this theorem together with the fact that the measures of the two arcs of
a circle sum to 360 degrees.
Given that line 𝐵𝐶 is a tangent
to the circle below, find the measure of angle 𝐴𝐵𝐶.
We will begin by marking the angle
we need to find on the diagram. We note that this is an angle of
tangency, as it lies between a tangent and a chord, in this case the tangent 𝐵𝐶
and the chord 𝐴𝐵. We know that the angle of tangency
is half of the measure of the arc formed on the same side. This means that the measure of the
angle 𝐴𝐵𝐶 is half the measure of the arc 𝐴𝐵 formed on the same side as
shown. The measure of the arc we have been
given, 190 degrees, is not on the same side. However, we can find the correct
arc measure by using the fact that the measures of two arcs of a circle sum to 360
degrees. The correct arc measure labeled two
𝜃 is equal to 360 degrees minus 190 degrees, which is equal to 170 degrees. Adding this to our diagram, we can
now calculate the angle of tangency. It is equal to a half of 170
degrees, which is equal to 85 degrees. This is the measure of angle
𝐴𝐵𝐶.
Let’s now consider one final
example that combines several of the theorems.
Given that the measure of angle
𝐸𝐶𝐷 is 54 degrees and the measure of angle 𝐹𝐵𝐷 is 78 degrees, find 𝑥 and
𝑦.
We will begin by adding the
information we have been given onto our diagram. We are told that angle 𝐸𝐶𝐷 is
equal to 54 degrees and angle 𝐹𝐵𝐷 is equal to 78 degrees. Since both of these angles are
angles of tangency, that is, they lie between a tangent and a chord, we will begin
by considering the alternate segment theorem. This states that the angles of
tangency are equal to the angles in the alternate segments. This means that the measure of
angle 𝐶𝐵𝐷 is equal to the measure of angle 𝐸𝐶𝐷. And from the information given,
these are both equal to 54 degrees. Likewise, the measure of angle
𝐵𝐶𝐷 is equal to the measure of angle 𝐹𝐵𝐷. And both of these are equal to 78
degrees.
We now have two of the three angles
of the inner triangle 𝐵𝐶𝐷. And since angles in a triangle sum
to 180 degrees, we have 𝑥 degrees plus 78 degrees plus 54 degrees is equal to 180
degrees. As all our measurements are in
degrees, we can rewrite the equation as shown. Subtracting 78 and 54 from both
sides, we are left with 𝑥 is equal to 48. The measure of angle 𝐵𝐷𝐶 is 48
degrees.
Having worked out 𝑥, we now need
to work out 𝑦. We begin by using the alternate
segment theorem once again or noting that angles on a straight line sum to 180
degrees. Using either of these, we find that
the measures of angle 𝐴𝐶𝐵 and 𝐴𝐵𝐶 are both equal to 48 degrees. This makes sense, since we recall
that two tangent segments meeting at a point have the same length and therefore form
an isosceles triangle when connected by a chord. Triangle 𝐴𝐵𝐶 is isosceles, where
the two equal angles are 48 degrees. This means that 𝑦 degrees plus 48
degrees plus 48 degrees is equal to 180 degrees. Once again, we can simplify the
equation. And subtracting 48 and 48 from both
sides gives us 𝑦 is equal to 84. The measure of angle 𝐵𝐴𝐶 is 84
degrees. And we can therefore conclude that
the values of 𝑥 and 𝑦 are 48 and 84, respectively.
Before summarizing the key points
from this video, we will briefly look at the converse of the alternate segment
theorem. This states that if a ray or line
segment meets with a chord of a circle on the outside of the circle and the angle it
makes with the chord is equal in measure to the angle in an alternate segment of the
circle, then the ray or line segment must be a tangent to the circle. If the angles are not equal, then
the ray or line segment is not tangent to the circle. This can be demonstrated in the two
diagrams as shown. In the first case, the measure of
angle 𝐴𝐶𝐷 is equal to the measure of the angle in the alternate segment. This means that line 𝐶𝐷 must be a
tangent, whereas in the second case, as the angles are not equal, 𝐶𝐷 cannot be a
tangent.
Let’s now recap what we learned
about angles of tangency in this video. The alternate segment theorem told
us that the angles of tangency 𝐴𝐶𝐷 and 𝐵𝐶𝐸 are equal to the angles in the
alternate segments 𝐴𝐵𝐶 and 𝐵𝐴𝐶, respectively. We also saw that for a circle of
center 𝑀, the angle of tangency 𝐴𝐵𝐶 is half of the central angle 𝐴𝑀𝐵. Finally, we saw that the angle of
tangency 𝐴𝐵𝐶 is half of the measure of the arc 𝐴𝐵 formed on the same side.
