### Video Transcript

In this video, we will learn how to
use cyclic quadrilateral properties to find missing angles and identify whether a
quadrilateral is cyclic or not. We will begin by recalling what is
meant by an inscribed angle.

An inscribed angle is the angle
made when two chords intersect on the circumference of a circle. This means that the vertex of the
angle lies on the circumference. We can use our understanding of
inscribed angles to define a cyclic quadrilateral. This is a four-sided polygon whose
vertices are inscribed on a circle. If we consider the cyclic
quadrilateral π΄π΅πΆπ·, we can join two vertices π΄ and πΆ to the center π in order
to create two radii π΄π and ππΆ. We can then label the angle
measures created at the center of the circle as π₯ degrees and π¦ degrees. Since angles about a point sum to
360 degrees, we have π₯ degrees plus π¦ degrees is equal to 360 degrees.

The inscribed angle theorem tells
us that an angle π inscribed in a circle is half of the central angle two π that
subtends the same arc on the circle, as shown. In other words, the angle of the
circumference is half the angle at the center. This means that the measure of the
angle at vertex π΅ is a half of π₯ degrees and the measure of the angle at vertex π·
is a half of π¦ degrees. We can then combine these three
equations. Firstly, we have the measure of
angle π΅ plus the measure of angle π· is equal to a half of π₯ degrees plus a half
of π¦ degrees. Factoring out a half on the
right-hand side gives us a half of π₯ degrees plus π¦ degrees. And since π₯ degrees plus π¦
degrees is 360 degrees, the measure of angle π΅ plus the measure of angle π· is
equal to a half of this, which is equal to 180 degrees. This means that the sum of this
pair of opposite angles is 180 degrees.

We can complete the same process to
demonstrate that the measure of angle π΄ and the measure of angle πΆ also sum to 180
degrees. And these two equations lead us to
the following property regarding opposite angles in a cyclic quadrilateral. The measures of the opposite angles
in a cyclic quadrilateral sum to 180 degrees. This means they are supplementary
angles. We can use this to calculate the
measures of missing angles in a cyclic quadrilateral. If the measure of angle π· was 75
degrees, we could calculate the measure of angle π΅ by subtracting this from 180,
giving us 105 degrees. In the same way, if angle π΄
measured 120 degrees, then the measure of angle πΆ would be 60 degrees as 180 minus
120 is 60. We will now consider an example
where the angles in the cyclic quadrilateral are given algebraically.

Given that the measure of angle
π΄ is equal to π¦ degrees, the measure of angle π΅ is equal to four π₯ minus
three degrees, and the measure of angle πΆ is equal to five π₯ degrees, find the
values of π₯ and π¦.

The figure shows a cyclic
quadrilateral π΄π΅πΆπ· such that the vertices of the quadrilateral lie on the
circumference of the circle. We recall that opposite angles
in a cyclic quadrilateral sum to 180 degrees. We are told in the question
that the measure of angle π΅ is four π₯ minus three degrees. And from the diagram, we see
that the measure of angle π· is 115 degrees. This means that four π₯ minus
three degrees plus 115 degrees is equal to 180. And since the angles are all
given in degrees, we can rewrite this as shown. Negative three plus 115 is
equal to 112. So the equation simplifies to
four π₯ plus 112 is equal to 180. We can then subtract 112 from
both sides such that four π₯ is equal to 68. And dividing through by four,
we have π₯ is equal to 17.

We are also told in the
question that the measure of angle π΄ is π¦ degrees and the measure of angle πΆ
is five π₯ degrees. This means that π¦ plus five π₯
equals 180. We have already calculated that
π₯ is equal to 17, so π¦ plus five multiplied by 17 is equal to 180. Multiplying five by 17 gives us
85. And subtracting this from both
sides, we have π¦ is equal to 180 minus 85. And this is equal to 95. The two solutions to this
question are π₯ equals 17 and π¦ is equal to 95.

Whilst it is not required in
this question, we could substitute these values back into the expressions for
the measures of angles π΄, π΅, and πΆ to calculate the missing angles. Angle π΄ is equal to 95
degrees. Four multiplied by 17 is
68. And subtracting three from this
gives us 65, so the measure of angle π΅ is 65 degrees. Finally, the measure of angle
πΆ is equal to 85 degrees. At this stage, as with any
quadrilateral, it is worth checking that all four angles sum to 360 degrees.

We will now consider how we can
extend the property of interior angles of a cyclic quadrilateral to the measure of
an exterior angle. Letβs begin by considering the
cyclic quadrilateral π΄π΅πΆπ·, where the measure of angle π΄ is π degrees and the
measure of angle πΆ is π degrees as shown. We know that π degrees plus π
degrees is equal to 180 degrees. And this can be rewritten as π
degrees is equal to 180 degrees minus π degrees. Letβs now consider an external
angle by extending the line segment π·πΆ to point πΈ in order to create an external
angle π΅πΆπΈ. If we label this angle β degrees
and since angles π΅πΆπ· and π΅πΆπΈ lie on a straight line, their measures will sum
to 180 degrees. This means that π degrees plus β
degrees is equal to 180 degrees.

Once again, we can rewrite this
equation as β degrees is equal to 180 degrees minus π degrees. Since the right-hand sides of our
two equations are equal, the left-hand sides must be equal and π degrees is equal
to β degrees. Replacing β with π on our diagram,
we see that angle π΅πΆπΈ is equal to π degrees. And this leads us to a general
property. An exterior angle of a cyclic
quadrilateral is equal to the interior angle at the opposite vertex. We will now apply this property to
an example.

Find the measure of angle
πΈπΆπΉ and the measure of angle π΄π΅πΉ.

In this question, weβre asked
to find the measure of two angles, firstly the measure of angle πΈπΆπΉ and
secondly the measure of angle π΄π΅πΉ. And in order to find these two
measures, weβll use two properties of cyclic quadrilaterals. Firstly, we recall that
opposite angles in a cyclic quadrilateral sum to 180 degrees. And secondly, exterior angles
of a cyclic quadrilateral are equal to the interior angle at the opposite
vertex. Using the second property, we
see that the measure of angle πΈπΆπΉ is equal to the measure of the angle at
vertex π΄ and is therefore equal to 80 degrees.

Using the same property, the
measure of the exterior angle π΄π΅πΉ is equal to the measure of the interior
angle at vertex π·, that is, the measure of angle π΄π·πΆ. Since angles on a straight line
sum to 180 degrees, we can calculate the measure of this angle by subtracting
104 degrees from 180 degrees. This is equal to 76
degrees. The measure of angle π΄π΅πΉ is
76 degrees. And we now have the two
solutions as required. Whilst we didnβt do so in this
question, we could have used the first property that opposite angles sum to 180
degrees to find the interior angles at vertices π΅ and πΆ first. We could then have used these
together with the fact that angles on a straight line sum to 180 degrees to find
the measures of πΈπΆπΉ and π΄π΅πΉ. Either way, we end up with two
answers of 80 degrees and 76 degrees.

We will now consider the converse
of these theorems. This states that a quadrilateral is
cyclic if we can demonstrate one of the following: either the opposite angles
measures are supplementary, i.e., they sum to 180 degrees, or an exterior angle is
equal to the interior angle at the opposite vertex. For example, since the opposite
angles in the quadrilateral drawn sum to 180 degrees, this must be a cyclic
quadrilateral. If, on the other hand, the two
angles did not sum to 180 degrees, the quadrilateral would not be cyclic and we
would not be able to draw a circle through all four vertices of the
quadrilateral. In the second diagram, since the
exterior angle is equal to the interior angle at the opposite vertex, the
quadrilateral π΄π΅πΆπ· is also cyclic. We will now look at an example
where we need to prove whether a quadrilateral is cyclic or not.

Is π΄π΅πΆπ· a cyclic
quadrilateral?

We begin by recalling that
there are two ways we can prove that a quadrilateral is cyclic: firstly, if the
opposite angles in the quadrilateral sum to 180 degrees and secondly if an
exterior angle is equal to the interior angle at the opposite vertex. It is the first of these we
will use in this question. If we can prove that the
measure of angle π΅ plus the measure of angle π· is equal to 180 degrees, then
the quadrilateral is cyclic. We can also do this with angles
π΄ and πΆ. We begin by noticing that
triangle π΄π·πΆ is isosceles. This means that the measure of
angle πΆπ΄π· is equal to the measure of angle π΄πΆπ·, which is equal to 53
degrees. Since angles in a triangle sum
to 180 degrees, the measure of angle π΄π·πΆ is equal to 180 degrees minus 53
degrees plus 53 degrees. This is therefore equal to 74
degrees.

We now have the measures of two
opposite angles in our quadrilateral. 106 plus 74 is equal to
180. So the measure of angle π΅ and
the measure of angle π· do sum to 180 degrees. And we can therefore conclude
that π΄π΅πΆπ· is a cyclic quadrilateral, and the correct answer is yes.

We will now summarize the key
points from this video. We saw in this video that a cyclic
quadrilateral is a four-sided polygon whose vertices are inscribed on a circle. We saw that in a cyclic
quadrilateral the measures of opposite angles are supplementary, i.e., they sum to
180 degrees, and also an exterior angle is equal to the interior angle at the
opposite vertex. Finally, we saw that the opposite
is also true. A quadrilateral is cyclic if we can
prove one of the following. The opposite angles sum to 180
degrees, or an exterior angle is equal to the interior angle at the opposite
vertex.