Lesson Explainer: Properties of Cyclic Quadrilaterals Mathematics

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, πœƒ, inscribed in a circle is half of the central angle, 2πœƒ, 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 𝐴𝐡𝐢𝐷. We can join two vertices, 𝐴 and 𝐢, to the center, 𝑂, to create two radii, 𝐴𝑂 and 𝑂𝐢. We can label the angle measures created at the center of the circle as π‘₯∘ and π‘¦βˆ˜.

Hence, we have π‘₯+𝑦=360(360).,π‘šβˆ π΅=12π‘₯()π‘šβˆ π·=12𝑦().∘∘∘∘∘∘theanglesaboutapointsumtoAlsoinscribedangletheoremandinscribedangletheorem

So, π‘šβˆ π΅+π‘šβˆ π·=12π‘₯+12𝑦=12(π‘₯+𝑦)=12(360)=180.∘∘∘∘∘

Hence, the sum of this pair of opposite angles is 180∘. We can complete the same process to demonstrate that π‘šβˆ π΄ and π‘šβˆ πΆ also sum to 180∘. 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 180∘.

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 π‘šβˆ π΅πΆπ·.

Answer

We can observe that the quadrilateral 𝐴𝐡𝐢𝐷 has all four vertices inscribed on the circumference of the circle. This means that 𝐴𝐡𝐢𝐷 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 180∘.

The angle measure that we need to determine, π‘šβˆ π΅πΆπ·, is opposite ∠𝐷𝐴𝐡.

We are given that π‘šβˆ π·π΄π΅=78∘ and we know that the measures of the two opposite angles sum to 180∘, so we calculate π‘šβˆ π·π΄π΅+π‘šβˆ π΅πΆπ·=18078+π‘šβˆ π΅πΆπ·=180π‘šβˆ π΅πΆπ·=180βˆ’78=102.∘∘∘∘∘∘

Thus, π‘šβˆ π΅πΆπ· is 102∘.

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 π‘šβˆ π΄=π‘¦βˆ˜, π‘šβˆ π΅=(4π‘₯βˆ’3)∘, and π‘šβˆ πΆ=5π‘₯∘, find the values of π‘₯ and 𝑦.

Answer

We can begin by adding the given angle measures to the diagram.

The quadrilateral 𝐴𝐡𝐢𝐷 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 180∘.

Using the pair of opposite angles ∠𝐡 and ∠𝐷, we can form an equation and substitute the values of the angles to give π‘šβˆ π΅+π‘šβˆ π·=180(4π‘₯βˆ’3)+115=1804π‘₯+112=1804π‘₯=180βˆ’1124π‘₯=68π‘₯=17.∘∘∘∘∘∘∘∘∘∘∘∘

We have now found the first unknown value, π‘₯=17. We can then use the remaining pair of opposite angles, ∠𝐴 and ∠𝐢, and solve the following equation: π‘šβˆ π΄+π‘šβˆ πΆ=180𝑦+5π‘₯=180.∘∘∘∘

As we calculated that π‘₯=17, we can substitute this into the equation and solve for 𝑦, giving 𝑦+5(17)=180𝑦+85=180𝑦=180βˆ’85=95.∘∘∘∘∘∘∘∘∘∘

As a useful check of our results, we can substitute our found values of π‘₯=17 and 𝑦=95 into the expression for each angle. This would give π‘šβˆ π΄=95,π‘šβˆ π΅=4(17)βˆ’3=65,π‘šβˆ πΆ=5(17)=85,π‘šβˆ π·=115.∘∘∘∘

We can confirm that the pair of opposite angles π‘šβˆ π΄+π‘šβˆ πΆ=180∘, and the other pair π‘šβˆ π΅+π‘šβˆ π·=180∘.

Furthermore, as with any quadrilateral, all four angle measures sum to 360∘. Thus, we have confirmed the result that π‘₯=17,𝑦=95.

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, 𝐴𝐡𝐢𝐷. We can denote π‘šβˆ π΄ as π‘“βˆ˜ and π‘šβˆ πΆ as π‘”βˆ˜.

Since 𝐴𝐡𝐢𝐷 is a cyclic quadrilateral, we know that 𝑓+𝑔=180.∘∘∘

Alternatively, we can also write that 𝑓=180βˆ’π‘”.∘∘∘

Let’s see what happens if we consider an external angle. We can extend the line segment 𝐷𝐢 to point 𝐸 to create an external angle, ∠𝐡𝐢𝐸.

Given that the angles 𝐡𝐢𝐷 and 𝐡𝐢𝐸 lie on a straight line, their measures will sum to 180∘. We can write that π‘šβˆ π΅πΆπ·+π‘šβˆ π΅πΆπΈ=180.∘

Given that π‘šβˆ π΅πΆπ·=π‘”βˆ˜, we can calculate 𝑔+π‘šβˆ π΅πΆπΈ=180π‘šβˆ π΅πΆπΈ=180βˆ’π‘”.∘∘∘∘

We have already established that there is another angle that is equivalent to 180βˆ’π‘”βˆ˜βˆ˜, since 𝑓=180βˆ’π‘”βˆ˜βˆ˜βˆ˜.

Hence, we have π‘šβˆ π΅πΆπΈ=𝑓.∘

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 π‘šβˆ πΈπΆπΉ and π‘šβˆ π΄π΅πΉ.

Answer

We observe that the quadrilateral 𝐴𝐡𝐢𝐷 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 ∠𝐸𝐢𝐹. The angle at the opposite vertex is that at vertex 𝐴, ∠𝐷𝐴𝐡. Since the exterior angle is equal to the interior opposite angle, we have π‘šβˆ πΈπΆπΉ=π‘šβˆ π·π΄π΅=80.∘

Next, we must calculate π‘šβˆ π΄π΅πΉ. This is an exterior angle, and we know that it must be equal to the interior angle at the opposite vertex, which is vertex 𝐷. Thus, π‘šβˆ π΄π΅πΉ=π‘šβˆ πΆπ·π΄.

We can mark these two congruent angles on the diagram.

We are not given the measure of angle 𝐢𝐷𝐴, but we can calculate it, given that the angle measures on a straight line sum to 180∘, and we are given π‘šβˆ πΆπ·πΊ=104∘. Hence, π‘šβˆ πΆπ·π΄+π‘šβˆ πΆπ·πΊ=180π‘šβˆ πΆπ·π΄+104=180π‘šβˆ πΆπ·π΄=180βˆ’104=76.∘∘∘∘∘∘

Then, since π‘šβˆ π΄π΅πΉ=π‘šβˆ πΆπ·π΄, we have π‘šβˆ π΄π΅πΉ=76.∘

We can then give the answers for the two required angles: π‘šβˆ πΈπΆπΉ=80,π‘šβˆ π΄π΅πΉ=76.∘∘

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, let’s say we have the the quadrilateral 𝐴𝐡𝐢𝐷 below.

We can calculate that π‘šβˆ π΄+π‘šβˆ πΆ=70+110=180.∘∘∘

Therefore, since these opposite pairs of angles sum to 180∘, 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 180∘. This is because as the polygon is a quadrilateral, we know that the total sum of the interior angles is 360∘. So, the angle sum of the two remaining (opposite) angles must be 360βˆ’180=180∘∘∘.

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.

Proving a Quadrilateral Is Cyclic

A quadrilateral is cyclic if we can demonstrate one of the following:

  • The opposite angles’ 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 𝐴𝐡𝐢𝐷 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, π‘šβˆ π΄π΅πΆ=82∘. We also have an exterior angle at 𝐢 split into two angles, ∠𝐷𝐢𝐹 and ∠𝐹𝐢𝐸. Notice that these two angles are marked as congruent; hence, π‘šβˆ πΉπΆπΈ=π‘šβˆ π·πΆπΉ=49∘.

We can calculate the measure of the interior angle at vertex 𝐷, π‘šβˆ π΄π·πΆ, by using the properties of the parallel lines, 𝐷𝐴 and 𝐢𝐡. We can write that π‘šβˆ π΄π·πΆ=π‘šβˆ π·πΆπΈ()=π‘šβˆ π·πΆπΉ+π‘šβˆ πΉπΆπΈ=49+49=98.alternateanglesareequal∘∘∘

If the quadrilateral is cyclic, then opposite angle measures sum to 180∘. Since we have 98+82=180,∘∘∘ therefore π‘šβˆ π΄π·πΆ=π‘šβˆ π΄π΅πΆ.

Thus, we have proved that the measures of the opposite angles sum to 180∘. Hence, we can answer the question: yes, the quadrilateral 𝐴𝐡𝐢𝐷 is cyclic.

Although not required, we can sketch the circle with the four vertices of 𝐴𝐡𝐢𝐷 inscribed on it.

Example 5: Using the Properties of Cyclic Quadrilaterals to Verify Whether a Given Quadrilateral Is Cyclic

Is 𝐴𝐡𝐢𝐷 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 180∘ or an exterior angle is equal to the interior angle at the opposite vertex.

Looking at the figure, we observe that △𝐴𝐢𝐷 is an isosceles triangle. Therefore, π‘šβˆ πΆπ΄π·=π‘šβˆ π·πΆπ΄=59.∘

Using the fact that the angle measures in a triangle sum to 180∘, and given that π‘šβˆ πΆπ΄π·=59∘, we can calculate π‘šβˆ π΄π·πΆ as π‘šβˆ π·πΆπ΄+π‘šβˆ πΆπ΄π·+π‘šβˆ π΄π·πΆ=18059+59+π‘šβˆ π΄π·πΆ=180118+π‘šβˆ π΄π·πΆ=180π‘šβˆ π΄π·πΆ=62.∘∘∘∘∘∘∘

If the quadrilateral is cyclic, then opposite angle measures sum to 180∘. However, 62+102β‰ 180;∘∘∘ thus, π‘šβˆ πΆπ·π΄+π‘šβˆ πΆπ΅π΄β‰ 180.∘

Therefore, the quadrilateral 𝐴𝐡𝐢𝐷 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 angles’ measures are supplementary.
    • An exterior angle is equal to the interior angle at the opposite vertex.

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