Lesson Video: Cyclic Quadrilaterals Mathematics

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.

11:55

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.

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