In this explainer, we will learn how to use cyclic quadrilateral properties to find missing angles and identify whether a quadrilateral is cyclic or not.
We can 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. The vertex of the angle lies on the circumference of the circle.
We use our understanding of inscribed angles to define a cyclic quadrilateral.
Definition: Cyclic Quadrilateral
A cyclic quadrilateral is a four-sided polygon whose vertices are inscribed on a circle.
Before we consider the properties of a cyclic quadrilateral, we recall an important theorem about inscribed angles and central angles (an angle at the center of a circle with end points on its circumference).
Definition: Inscribed Angle Theorem
An angle, , inscribed in a circle is half of the central angle, , that subtends the same arc on the circle. In other words, the angle at the circumference is half the angle at the center.
Let us now consider the cyclic quadrilateral . We can join two vertices, and , to the center, , to create two radii, and . We can label the angle measures created at the center of the circle as and .
Hence, we have
So,
Hence, the sum of this pair of opposite angles is . We can complete the same process to demonstrate that and also sum to . Therefore, in a cyclic quadrilateral, we have proven the following property.
Definition: Opposite Angles in a Cyclic Quadrilateral
The measures of the opposite angles in a cyclic quadrilateral are supplementary; that is, they add to .
We will see some examples of how we can use this property to find missing angles in a cyclic quadrilateral.
Example 1: Finding the Measure of an Angle in a Cyclic Quadrilateral given the Measure of the Opposite Angle
Determine .
Answer
We can observe that the quadrilateral has all four vertices inscribed on the circumference of the circle. This means that is a cyclic quadrilateral, and we can use the angle properties of a cyclic quadrilateral to help us find the unknown angle. The measures of opposite angles in a quadrilateral sum to .
The angle measure that we need to determine, , is opposite .
We are given that and we know that the measures of the two opposite angles sum to , so we calculate
Thus, is .
We can also apply our understanding of the angles in a cyclic quadrilateral to geometric problems where the angle measures are given algebraically. Let us see how in the following example.
Example 2: Using the Properties of Cyclic Quadrilaterals to Find Unknown Values
Given that , , and , find the values of and .
Answer
We can begin by adding the given angle measures to the diagram.
The quadrilateral is a cyclic quadrilateral, as all four vertices are inscribed on the circle. We recall that in a cyclic quadrilateral the measures of opposite angles are supplementary: they add to .
Using the pair of opposite angles and , we can form an equation and substitute the values of the angles to give
We have now found the first unknown value, . We can then use the remaining pair of opposite angles, and , and solve the following equation:
As we calculated that , we can substitute this into the equation and solve for , giving
As a useful check of our results, we can substitute our found values of and into the expression for each angle. This would give
We can confirm that the pair of opposite angles , and the other pair .
Furthermore, as with any quadrilateral, all four angle measures sum to . Thus, we have confirmed the result that
We will now see how we can extend the property of interior angles of cyclic quadrilaterals to the measure of an exterior angle.
We can consider the following quadrilateral, . We can denote as and as .
Since is a cyclic quadrilateral, we know that
Alternatively, we can also write that
Letβs see what happens if we consider an external angle. We can extend the line segment to point to create an external angle, .
Given that the angles and lie on a straight line, their measures will sum to . We can write that
Given that , we can calculate
We have already established that there is another angle that is equivalent to , since .
Hence, we have
We can repeat this method to demonstrate that for every angle in a cyclic quadrilateral, an exterior angle is equal to the interior angle at the opposite vertex. We can note this property below.
Definition: Exterior Angles in a Cyclic Quadrilateral
An exterior angle of a cyclic quadrilateral is equal to the interior angle at the opposite vertex.
We will now see how we can apply this property to find missing angles in a geometrical problem involving a cyclic quadrilateral.
Example 3: Using the Properties of Cyclic Quadrilaterals to Solve Problems
Find the and .
Answer
We observe that the quadrilateral is a cyclic quadrilateral since all four vertices are inscribed on the circle. There are two important angle properties in cyclic quadrilaterals that will be useful in this problem. In a cyclic quadrilateral, opposite angle measures are supplementary. As this figure also includes external angles, we should also remember that an exterior angle of a cyclic quadrilateral is equal to the interior angle at the opposite vertex.
The first angle to calculate is . The angle at the opposite vertex is that at vertex , . Since the exterior angle is equal to the interior opposite angle, we have
Next, we must calculate . This is an exterior angle, and we know that it must be equal to the interior angle at the opposite vertex, which is vertex . Thus,
We can mark these two pairs of congruent angles on the diagram.
We are not given the measure of angle , but we can calculate it, given that the angle measures on a straight line sum to , and we are given . Hence,
Then, since we have
We can then give the answers for the two required angles:
So far, we have used the property that a cyclic quadrilateral has opposite angle measures that are supplementary. However, the converse of this theorem is also true; that is, a quadrilateral with opposite angles that are supplementary must be a cyclic quadrilateral. This is particularly useful if we wish to prove that a given quadrilateral is cyclic, and therefore demonstrate that all vertices could be inscribed upon a single circle.
For example, letβs say we have the quadrilateral below.
We can calculate that
Therefore, since these opposite pairs of angles sum to , the quadrilateral is cyclic, and so all four vertices can be inscribed onto a circle.
Notice that we do not need to prove that the other pair of opposite angles also sum to . This is because as the polygon is a quadrilateral, we know that the total sum of the interior angles is . So, the angle sum of the two remaining (opposite) angles must be .
The same is true for the other angle property in cyclic properties we have seen: an exterior angle of a cyclic quadrilateral is equal to the interior angle at the opposite vertex. Since this property is an extension of the opposite angle property, the converse of this theorem is also true.
Property: Proving a Quadrilateral Is Cyclic
A quadrilateral is cyclic if we can demonstrate one of the following:
- The opposite anglesβ measures are supplementary.
- An exterior angle is equal to the interior angle at the opposite vertex.
We will now see how we can apply these rules to establish if a quadrilateral is cyclic in the following two examples.
Example 4: Determining Whether a Given Quadrilateral Is Cyclic
Is the quadrilateral cyclic?
Answer
We can recall that a cyclic quadrilateral is a four-sided polygon whose vertices are inscribed on a circle. One way to prove if a quadrilateral is cyclic is to establish if an angle property of cyclic quadrilaterals is satisfied. If it is, then the quadrilateral is cyclic.
We can observe, in the figure we are given, one interior angle, . We also have an exterior angle at split into two angles, and . Notice that these two angles are marked as congruent; hence, .
We can calculate the measure of the interior angle at vertex , , by using the properties of the parallel lines, and . We can write that
If the quadrilateral is cyclic, then opposite angle measures sum to . Since we have therefore
Thus, we have proved that the measures of the opposite angles sum to . Hence, we can answer the question: yes, the quadrilateral is cyclic.
Although not required, we can sketch the circle with the four vertices of inscribed on it.
Example 5: Using the Properties of Cyclic Quadrilaterals to Verify Whether a Given Quadrilateral Is Cyclic
Is a cyclic quadrilateral?
Answer
We recall that a quadrilateral is cyclic if all four vertices can be inscribed on a circle. We can prove a quadrilateral is cyclic if either of the following properties can be demonstrated: a pair of opposite angle measures sum to or an exterior angle is equal to the interior angle at the opposite vertex.
Looking at the figure, we observe that is an isosceles triangle. Therefore,
Using the fact that the angle measures in a triangle sum to , and given that , we can calculate as
If the quadrilateral is cyclic, then opposite angle measures sum to . However, thus,
Therefore, the quadrilateral is not cyclic, so the answer to the question is no.
We can summarize the key points from this explainer.
Key Points
- A cyclic quadrilateral is a four-sided polygon whose vertices are inscribed on a circle.
- In a cyclic quadrilateral,
- the measures of opposite angles are supplementary,
- an exterior angle is equal to the interior angle at the opposite vertex.
- A quadrilateral is cyclic if we can prove one of the following:
- The opposite anglesβ measures are supplementary.
- An exterior angle is equal to the interior angle at the opposite vertex.
