Lesson Video: Properties of Cyclic Quadrilaterals | Nagwa Lesson Video: Properties of Cyclic Quadrilaterals | Nagwa

Lesson Video: Properties of Cyclic Quadrilaterals Mathematics • Third Year of Preparatory School

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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.

16:00

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 𝐴𝐡𝐢𝐷, we can join two vertices 𝐴 and 𝐢 to the center 𝑂 in order to create two radii 𝐴𝑂 and 𝑂𝐢. We can then label the angle measures created at the center of the circle as π‘₯ degrees and 𝑦 degrees. Since angles about a point sum to 360 degrees, we have π‘₯ degrees plus 𝑦 degrees is equal to 360 degrees.

The inscribed angle theorem tells us that an angle πœƒ inscribed in a circle is half of the central angle two πœƒ 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 𝐡 is a half of π‘₯ degrees and the measure of the angle at vertex 𝐷 is a half of 𝑦 degrees. We can then combine these three equations. Firstly, we have the measure of angle 𝐡 plus the measure of angle 𝐷 is equal to a half of π‘₯ degrees plus a half of 𝑦 degrees. Factoring out a half on the right-hand side gives us a half of π‘₯ degrees plus 𝑦 degrees. And since π‘₯ degrees plus 𝑦 degrees is 360 degrees, the measure of angle 𝐡 plus the measure of angle 𝐷 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 𝐴 and the measure of angle 𝐢 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 𝐷 was 75 degrees, we could calculate the measure of angle 𝐡 by subtracting this from 180, giving us 105 degrees. In the same way, if angle 𝐴 measured 120 degrees, then the measure of angle 𝐢 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 𝐴 is equal to 𝑦 degrees, the measure of angle 𝐡 is equal to four π‘₯ minus three degrees, and the measure of angle 𝐢 is equal to five π‘₯ degrees, find the values of π‘₯ and 𝑦.

The figure shows a cyclic quadrilateral 𝐴𝐡𝐢𝐷 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 𝐡 is four π‘₯ minus three degrees. And from the diagram, we see that the measure of angle 𝐷 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 𝐴 is 𝑦 degrees and the measure of angle 𝐢 is five π‘₯ degrees. This means that 𝑦 plus five π‘₯ equals 180. We have already calculated that π‘₯ is equal to 17, so 𝑦 plus five multiplied by 17 is equal to 180. Multiplying five by 17 gives us 85. And subtracting this from both sides, we have 𝑦 is equal to 180 minus 85. And this is equal to 95. The two solutions to this question are π‘₯ equals 17 and 𝑦 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 𝐴, 𝐡, and 𝐢 to calculate the missing angles. Angle 𝐴 is equal to 95 degrees. Four multiplied by 17 is 68. And subtracting three from this gives us 65, so the measure of angle 𝐡 is 65 degrees. Finally, the measure of angle 𝐢 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. Let’s begin by considering the cyclic quadrilateral 𝐴𝐡𝐢𝐷, where the measure of angle 𝐴 is 𝑓 degrees and the measure of angle 𝐢 is 𝑔 degrees as shown. We know that 𝑓 degrees plus 𝑔 degrees is equal to 180 degrees. And this can be rewritten as 𝑓 degrees is equal to 180 degrees minus 𝑔 degrees. Let’s now consider an external angle by extending the line segment 𝐷𝐢 to point 𝐸 in order to create an external angle 𝐡𝐢𝐸. If we label this angle β„Ž degrees and since angles 𝐡𝐢𝐷 and 𝐡𝐢𝐸 lie on a straight line, their measures will sum to 180 degrees. This means that 𝑔 degrees plus β„Ž degrees is equal to 180 degrees.

Once again, we can rewrite this equation as β„Ž degrees is equal to 180 degrees minus 𝑔 degrees. Since the right-hand sides of our two equations are equal, the left-hand sides must be equal and 𝑓 degrees is equal to β„Ž degrees. Replacing β„Ž with 𝑓 on our diagram, we see that angle 𝐡𝐢𝐸 is equal to 𝑓 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 𝐸𝐢𝐹 and the measure of angle 𝐴𝐡𝐹.

In this question, we’re asked to find the measure of two angles, firstly the measure of angle 𝐸𝐢𝐹 and secondly the measure of angle 𝐴𝐡𝐹. And in order to find these two measures, we’ll 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 𝐸𝐢𝐹 is equal to the measure of the angle at vertex 𝐴 and is therefore equal to 80 degrees.

Using the same property, the measure of the exterior angle 𝐴𝐡𝐹 is equal to the measure of the interior angle at vertex 𝐷, that is, the measure of angle 𝐴𝐷𝐢. 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 𝐴𝐡𝐹 is 76 degrees. And we now have the two solutions as required. Whilst we didn’t 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 𝐡 and 𝐢 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 𝐸𝐢𝐹 and 𝐴𝐡𝐹. 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 𝐴𝐡𝐢𝐷 is also cyclic. We will now look at an example where we need to prove whether a quadrilateral is cyclic or not.

Is 𝐴𝐡𝐢𝐷 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 𝐡 plus the measure of angle 𝐷 is equal to 180 degrees, then the quadrilateral is cyclic. We can also do this with angles 𝐴 and 𝐢. We begin by noticing that triangle 𝐴𝐷𝐢 is isosceles. This means that the measure of angle 𝐢𝐴𝐷 is equal to the measure of angle 𝐴𝐢𝐷, which is equal to 53 degrees. Since angles in a triangle sum to 180 degrees, the measure of angle 𝐴𝐷𝐢 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 𝐡 and the measure of angle 𝐷 do sum to 180 degrees. And we can therefore conclude that 𝐴𝐡𝐢𝐷 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.

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