In this explainer, we will learn how to identify the angle of tangency in a circle
and find its measure using the measure
of its subtended arc, inscribed angle, or central angle subtended by the same arc.
Let us begin by recalling that a tangent line to a circle is a line that
intersects the circle at just one point, as shown below.
We note that the line segment from the point of intersection
to the center of the circle is a
radius of the circle. Furthermore, this radius is perpendicular
(i.e., at 90 degrees)
to the tangent line.
In this explainer, we want to discuss angles of tangency. Consider a tangent to a
circle that meets with a chord of the circle (i.e., a line segment on the inside of
the circle) at the point .
The angle between the chord and the tangent is known as
an angle of tangency. Calculating this angle can be done with the help of
several theorems and observations that we will discuss over the course of this
explainer.
It will also be important to keep in mind some of the properties of triangles. Isosceles and equilateral triangles are triangles that have two or three equal
sides, respectively, as shown below.
It is important to note that because the radii of a circle all have the same
length, two radii can form the sides of an isosceles triangle as shown
below.
This can be useful to know, since it tells us that the measures of the angles at
and are equal.
Additionally, let us review the inscribed angle theorem (also called the
central angle theorem), which is crucial for upcoming calculations that deal with
angles within a circle.
Recall that two points on a circle,
and , divide a circle into two arcs: a major arc and
a minor arc (when the arcs are the same length, they divide the circle into
two semicircular arcs). We can also form an inscribed angle with any point
on the major arc as shown.
Note that sometimes, the major arc is referred to as the subtended arc
and the minor arc the intercepted arc. Then, we have the following
theorem.
Theorem: Inscribed Angle Theorem
Let and be two points on a circle,
be the center of the circle, and be any
point on the major arc. Then, the measure of the central angle is double the measure of the
inscribed angle, as shown.
Another way to phrase this theorem 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 us learn a new theorem that deals with angles of
tangency in a circle.
Theorem: Alternate Segment Theorem
Let and be two points on a circle and
be the point where a tangent (passing through
, , and ) intersects the
circle. Then, the angles of tangency
and are equal to the angles in the
alternate segments and
respectively. This is shown below.
Let us prove this theorem. To start with, recall that a tangent of the circle at
point forms a right angle with the radius
at the point of intersection.
Now, we begin by considering one of the angles of tangency
that we label as
below.
Because the tangent and the radius form a right angle, we know that
. Therefore, . We also know that the inner triangle is an isosceles
triangle because two of its sides are radii and are therefore equal. So,
as well.
Since a triangle always has all of its angles adding up to
, we have
Rearranging, we have
This gives us that the measure of the central angle between
and is . Finally,
using the inscribed angle theorem, we can conclude that the measure of
is half the central angle
, since it is the inscribed angle of and
. Thus, .
We note that the same method applies to the other angle of tangency
, since we can just repeat the process with the
chord . Thus, we have proved the alternate segment
theorem.
Let us see an example where we can put this theorem directly to use.
Example 1: Finding the Measure of an Angle of Tangency given the Measure of the
Inscribed Angle Subtended by the Same Arc
Given that is a tangent to the
circle, find .
Answer
Let us first of all mark the angle we want to find on the diagram with
.
For any question where we need to find the angle of tangency, we need to
ask ourselves whether we can use the alternate segment theorem
to help us.
Now, we know that is a tangent to the
circle at , that , ,
and are three points on the circle, and that the angle we
want to find is the angle of tangency . Therefore, we can use the alternate segment theorem to find this angle. That is,
Since is the angle in the alternate
segment to , we directly use the theorem
to find that .
We have seen how the alternate segment theorem can be used directly to find
inscribed angles given the angle of tangency and vice versa, but we can also use
other aspects of the theorem to help solve different problems. For instance, let us
consider how angles of tangency relate to central angles.
Corollary: Angles of Tangency and Central Angles
Let be a point on a circle of center
and be the point where a tangent
(passing through and ) intersects the
circle. Then, the angle of tangency is half
of the central angle . This is shown
below.
Another way to phrase this is that the angle of tangency is half the central angle
subtended by the same arc (i.e., the arc ).
Note that as this corollary is something we demonstrated during the proof of the
alternate segment theorem, we therefore do not need to prove it again. Let us
consider an example where we can use this theorem directly to find the angle of
tangency.
Example 2: Finding the Measure of an Angle of Tangency given the Measure of the
Central Angle Subtended by the Same Arc
Find .
Answer
Let us begin by marking the angle we have been asked to find on the
diagram:
We can see that we have been asked to find the measure of an angle of
tangency . Note that the angle
on the diagram is a right angle
because it is marked with a square, thus telling us its measure is
. We also note that
is the central angle subtended by the
same arc (i.e., ) as the angle of
tangency.
Recall that the measure of the angle of tangency
is equal to half the measure of
the central angle subtended by the same arc, which is
. Thus, we have
So far, 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.
Recall that the measure of an arc is the angle that the arc makes at the center
of the circle. For instance, consider the diagram below.
Here, we can see that the measure of the major arc of the circle created by the
two points and (denoted by
on the outside of the circle) is equivalent to the measure
of the angle between the radii formed by and
in the inside of the circle.
Recall that we already have a corollary that relates the central angle of a circle
to the angle of tangency. By using the equivalency between the central angle and
the arc measure, we can extend this corollary to apply to the arc of a circle. This
equivalency can be seen in the following diagram.
In other words, since both the measure of the central angle and the arc measure
are , they are both twice the angle of tangency. Thus, we
have the following corollary.
Corollary: Angles of Tangency and Arc Measures
Let be a point on a circle and
be the point where a tangent (passing through and
) intersects the circle. Then, the angle of tangency
is half of the measure of the arc
formed on the same side. This is shown
below.
It is important to realize which arc refers to,
since there are two possibilities: the major arc and the minor arc
(also known as the subtended and intercepted arcs). In the diagram above, the minor
arc was used, since it is on the same side as the angle of tangency. However, the
opposite would be true if we considered an obtuse angle, as shown below.
Here, is now a major arc of the circle. It is
also important to note that if we are given the opposite arc to the one we need,
we can make use of the fact that the measures of the two arcs of a circle add up to
. So, it is always
possible to find the major arc if we are given the minor arc,
or vice versa.
Let us consider an example where we can use this theorem to calculate an angle of
tangency using the measure of an arc.
Example 3: Finding the Measure of an Angle of Tangency Using Arc Measure
Given that is a tangent to the circle
below, find .
Answer
Let us begin by marking the angle we need to find on the diagram.
Recall that the angle of tangency
is half of the measure of the arc
formed on the same side. In this example, we can see that the arc measure
we have been given (i.e., )
is not on the same side. However, we can find the correct arc measure by
using the fact that the measures of the two arcs of a circle have to sum to
. Thus, the correct arc measure is equal to
Let us mark this on the diagram.
Now, we can use the fact that the angle of tangency is half the measure
of the arc on the same side to get
One other type of question involving angles of tangency has a point outside of
the circle that goes through two different tangent lines. Let us consider an
example of this.
Example 4: Finding the Measures of Two Inscribed Angles given the Measures of Angles
of Tangency Subtended by the Same Arcs to Find Other Unknown Angles
Given that
and ,
find and .
Answer
Let us begin by putting the information we have been given,
and , into the
diagram.
To start with, let us try to find . As this is a
question involving angles of tangency, we ask ourselves whether the
alternate segment theorem can be used to help us. We can see that as there
are two angles of tangency and three points on the circle, we can use the
theorem twice in the inner triangle to find missing angles. This is shown
below.
Now, we notice that we have two of the three angles of the inner triangle. Since the angles of a triangle add up to
, we have
Having found , we now want to find ,
which means finding the angles of the triangle that is in. To find the other angles, we can make use of the fact that the angles of a
straight line have to add up to
. Thus, using the
exact calculation as before, we find that the remaining angles must be
as well, giving us
the following.
Recalling that two tangent segments meeting at a point have the same
length and thus form an isosceles triangle when connected by a chord,
we can confirm that these two angles being equal is consistent with this
rule. Finally, we can calculate using the sum of
angles in a triangle.
So, all in all, we have
and .
Up until now, we have used the alternate segment theorem and variations of it to
find the angle that a tangent makes with a chord in a circle. Just as it is
possible to use the theorem to find angles in this way, we can also use the
converse to prove that a given ray or segment is a tangent to the circle if the
corresponding angles match up. Formally, we have the following corollary.
Corollary: Converse of Alternate Segment Theorem
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 that ray or line segment
must be a tangent to the circle.
If the angles are not equal, then that ray or line segment is not tangent to
the circle.
Let us illustrate what this means specifically. Suppose we have a ray
and we are given the angle that it makes with the
chord . Then, we have two possibilities,
as shown below.
In the first case, the measure of
is equal to that of the measure of the angle in the alternate segment
; consequently,
must be a tangent. In the second case,
the angles are not equal, since ; hence,
cannot be a tangent.
One particular situation we may have to use this corollary is with circumcircles. Recall that if we are given a triangle, there exists exactly one circle that passes
through all the vertices of the triangle. We call this circle a
circumcircle.
Sometimes, we may be asked questions regarding tangents to the circumcircle of a
triangle. For instance, we may have to verify that a ray is a tangent to a
circumcircle by measuring its angle with a chord and confirming that it satisfies
the properties of an angle of tangency. Let us explore this idea in the following
example.
Example 5: Finding the Angle of Tangency given That a Line Is Tangent to the
Circumcircle of a Triangle
In the given figure, if , which of the
following is a tangent to the circle that passes through the vertices of
the triangle ?
Answer
To best understand this question, let us annotate on the diagram the
information we are being asked to find. Specifically, we need to consider the
circle that passes through the vertices of the triangle
(i.e., a circumcircle). Let us highlight this
triangle.
Although we have not yet drawn a circumcircle around ,
we can immediately see that
and cannot be tangential to the
circle, since they are extensions of the sides of the triangle
(in other words, they are secants). It remains to be seen whether
,
, or
is tangent to the circle.
In order to verify these other options, we need to investigate the
surrounding angles and find whether it makes sense for them to be tangents. In particular, we can use the converse of the alternate segment theorem to
prove in each case whether they are a tangent or not.
Let us begin by considering
. Recall that in the problem
description, it has been given to us that . In particular, this means that the corresponding arcs are not equal in
length. Let us consider the arcs
and . We note that the angle
is an inscribed angle of
and
is an inscribed angle of
. We highlight these arcs and the
corresponding inscribed angles below.
Recall that the measure of an inscribed angle subtended by an arc is half
the measure of the arc. Since arcs and
have different lengths, this means
and
are not equal.
Now, let us recall the converse of the alternate segment theorem: if the
angle between and
(which is a chord of the
circumcircle) is not equal to the angle in the alternate segment
(i.e., ), then
cannot be a tangent to this circle. Thus, is not a tangent.
Next, let us look at . Consider the triangle
. We can see that
is an exterior angle to this triangle,
which means that it is equal to the sum of the remote interior angles
(by the exterior angle theorem). In other words,
We highlight this below.
Now, since is a tangent to the larger
circle, we can use the alternate segment theorem on it. In particular, if we
consider the triangle , we can see that
We indicate this on the diagram too.
However, now we can use the converse of the alternate segment theorem in
the triangle : substituting equation
(2) into (1), we have
Since , this means that
So, the two angles cannot be equal, which means
cannot be tangent to the circle passing
through triangle .
Finally, let us consider . To begin with, let us use the information given in the question. We can see
that and
are parallel lines, since they have
been marked with double arrows. This means that the alternate angles
and
must be equal. We highlight this below.
Now, we can see that is a tangent to
the larger circle and that there are multiple triangles inside this circle,
so we can use the alternate segment theorem here. In particular, consider the
triangle . We can see that the angle of tangency
is equal to the angle in the alternate
segment, , as shown.
Now, returning to the triangle , we sketch the
circumcircle going round it and highlight the angles
and :
Let us use the converse of the alternate segment theorem once more. Specifically, since the angles and
have been shown to be equal, it must
therefore be the case that is a
tangent to the circle.
Thus, the answer is E: .
Let us finish off by recapping the alternate segment theorem and the places we can
use it.
Key Points
Let and be two points on a circle
and be the point where a tangent (passing through
, , and ) intersects
the circle. Then, the angles of tangency
and
are equal to the angles in the alternate
segments and
respectively.
As an extension of the above, for a circle of center ,
the angle of tangency is half of the
central angle .
Additionally, the angle of tangency
is half of the measure of the arc
formed on the same side.
Conversely, 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 that ray or line segment
must be a tangent to the circle.
On the other hand, if the angles are not equal, then that ray or line segment is
not tangent to the circle.
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