Video Transcript
In this video, we will use the
properties of cyclic quadrilaterals to find missing angles and also to identify
whether a quadrilateral is cyclic or not.
We will begin by defining what we
mean by a cyclic quadrilateral. A quadrilateral is a four-sided
shape. And a cyclic quadrilateral is a
quadrilateral whose vertices all lie on the circumference of a single circle. In our diagram, the vertices 𝐴,
𝐵, 𝐶, and 𝐷 lie on the circumference of this circle. The four angles inside any
quadrilateral sum to 360 degrees. This means that angle 𝐴 plus angle
𝐵 plus angle 𝐶 plus angle 𝐷 equals 360 degrees.
The opposite angles in a cyclic
quadrilateral sum to 180 degrees. In our diagram, angle 𝐴 plus angle
𝐶 is equal to 180 degrees, and angle 𝐵 plus angle 𝐷 is also equal to 180
degrees. As 180 plus 180 is equal to 360,
this satisfies the first property.
We will now look at some questions
where we will determine whether a quadrilateral is cyclic or not.
Is 𝐴𝐵𝐶𝐷 a cyclic
quadrilateral?
In any cyclic quadrilateral, we
know that the opposite angles sum to 180 degrees. As angles 𝐴 and 𝐶 are
opposite, this would mean that angle 𝐴 plus angle 𝐶 must be equal to 180
degrees. 79 degrees plus 62 degrees is
equal to 141 degrees. This means that angle 𝐴 plus
angle 𝐶 is not equal to 180 degrees. We can therefore conclude that
the answer is no, 𝐴𝐵𝐶𝐷 is not a cyclic quadrilateral, as the sum of angles
𝐴 and 𝐶 is not 180 degrees.
We will now look at a second
question of a similar type.
Is 𝐴𝐵𝐶𝐷 a cyclic
quadrilateral?
We know that the opposite
angles in a cyclic quadrilateral sum to 180 degrees. In this question, we will
consider the opposite angles 𝐵 and 𝐷. It follows that if these two
angles sum to 180 degrees, then angles 𝐴 and 𝐶 must also sum to 180 degrees,
as the four angles inside a quadrilateral sum to 360 degrees.
We can begin this question by
recalling that the three angles inside any triangle sum to 180 degrees. This means that angle 𝐵 plus
48 degrees plus 29 degrees must equal 180 degrees. 48 plus 29 is equal to 77. Subtracting 77 degrees from
both sides of this equation gives us angle 𝐵 is equal to 103 degrees. We can now go back to our
statement about a cyclic quadrilateral. We know that angle 𝐵 is equal
to 103 degrees and angle 𝐷 is equal to 77 degrees. These two indeed sum to 180
degrees.
As previously mentioned, if
angle 𝐵 plus angle 𝐷 sum to 180 degrees, then angle 𝐴 plus angle 𝐶 must also
sum to 180 degrees. This is because the four angles
inside the quadrilateral 𝐴𝐵𝐶𝐷 must sum to 360 degrees. We can therefore conclude that
yes, 𝐴𝐵𝐶𝐷 is a cyclic quadrilateral.
Our next few questions involve
calculating missing angles in cyclic quadrilaterals.
Determine the measure of angle
𝐵𝐶𝐷.
The angle 𝐵𝐶𝐷 is the one
shown in the diagram. The four vertices of our
quadrilateral 𝐴𝐵𝐶𝐷 lie on the circumference of the circle. This means that our
quadrilateral is cyclic. We know that the opposite
angles in a cyclic quadrilateral sum to 180 degrees. This means that the sum of
angle 𝐴 and angle 𝐶 is 180 degrees. This is also true for angles 𝐵
and 𝐷.
We know that angle 𝐴 is equal
to 78 degrees. Subtracting 78 degrees from
both sides of this equation gives us angle 𝐶 is equal to 180 degrees minus 78
degrees. This is equal to 102
degrees. The measure of angle 𝐵𝐶𝐷 in
the quadrilateral is 102 degrees.
We will now look at a slightly more
complicated question.
Given that 𝐴𝐵𝐶𝐷 is a cyclic
quadrilateral, find the measure of angle 𝐵𝐴𝐶.
Angle 𝐵𝐴𝐶 is shown on the
diagram labeled 𝑥. The marks on lines 𝐴𝐵 and
𝐵𝐶 indicate that these two sides are equal in length. This means that triangle 𝐴𝐵𝐶
is isosceles. An isosceles triangle has two
equal angles. In this case, angle 𝐵𝐴𝐶 is
equal to angle 𝐵𝐶𝐴. The opposite angles in a cyclic
quadrilateral sum to 180 degrees. If we let angle 𝐴𝐵𝐶 equal
the letter 𝑦, we know that this angle plus angle 𝐴𝐷𝐶 must sum to 180
degrees. 𝑦 plus 61 degrees is equal to
180 degrees. Subtracting 61 degrees from
both sides of this equation gives us 𝑦 is equal to 119 degrees.
We know that the angles in any
triangle sum to 180 degrees. This means that 𝑥 plus 𝑥 plus
119 degrees must equal 180 degrees. Simplifying the left-hand side
of the equation gives us two 𝑥 plus 119 degrees. When we subtract this from both
sides of the new equation, we get two 𝑥 is equal to 61 degrees. Finally, dividing both sides of
the equation by two gives us 𝑥 is equal to 30.5 degrees. We can therefore conclude that
the measure of angle 𝐵𝐴𝐶 is 30.5 degrees.
We will now look at a cyclic
quadrilateral where there are two unknowns that we need to calculate.
Given that the measure of angle
𝐵𝐴𝐷 is 𝑥 plus 34 degrees, find the values of 𝑥 and 𝑦.
We’re told in the question that
angle 𝐵𝐴𝐷 is equal to 𝑥 plus 34 degrees. We also notice from the diagram
that triangle 𝐵𝐴𝐸 is isosceles, as the lengths 𝐵𝐴 and 𝐵𝐸 are equal. This means that angle 𝐵𝐴𝐸 is
equal to angle 𝐵𝐸𝐴, which is equal to 51 degrees. We know that the angles in a
triangle sum to 180 degrees. This means that the angle
𝐴𝐵𝐸 plus 51 degrees plus 51 degrees is equal to 180 degrees. 51 plus 51 is equal to 102. And subtracting this from 180
gives us angle 𝐴𝐵𝐸 is 78 degrees.
We also know that angles on a
straight line sum to 180 degrees. This means that we can
calculate the angle 𝐴𝐵𝐶 inside the cyclic quadrilateral by subtracting 78
degrees from 180 degrees. Angle 𝐴𝐵𝐶 is equal to 102
degrees. The quadrilateral 𝐴𝐵𝐶𝐷 is
cyclic as all four vertices lie on the circumference of a circle. We know that opposite angles in
any cyclic quadrilateral sum to 180 degrees. This means that 𝑥 plus 102 is
equal to 180 and 𝑦 plus 𝑥 plus 34 is also equal to 180.
As just mentioned by looking at
the angles at vertices 𝐵 and 𝐷, we have the equation 𝑥 plus 102 degrees is
equal to 180 degrees. Subtracting 102 from both sides
of this equation gives us 𝑥 is equal to 78 degrees. The angles at vertices 𝐴 and
𝐶 will also sum to 180 degrees. 78 plus 34 is equal to 112. Subtracting this from both
sides of the equation gives us 𝑦 is equal to 68 degrees. The values of 𝑥 and 𝑦,
respectively, are 78 degrees and 68 degrees. We could check this by adding
the four angles inside the quadrilateral, which must sum to 360 degrees.
We will now look at some of the key
points from this video. A cyclic quadrilateral has all four
vertices on the circumference of a circle. The opposite angles in a cyclic
quadrilateral sum to 180 degrees. This is one of our key circle
theorems that will be tested on in examinations. This fact can be used alongside
other angle properties that we know, as seen in this video. These can include angles in a
triangle and angles on a straight line. Both of these sum to 180
degrees. The property can be seen in the
diagram as shown, where angle 𝐴 plus angle 𝐶 equal 180 degrees and angle 𝐵 plus
angle 𝐷 also equals 180 degrees. The sum of all four angles in any
quadrilateral equals 360 degrees.
