Lesson Video: Proving Cyclic Quadrilaterals | Nagwa Lesson Video: Proving Cyclic Quadrilaterals | Nagwa

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

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In this video, we will learn how to prove that a quadrilateral is cyclic using the angles resulting from its diagonals.

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### Video Transcript

In this video, we will learn how to prove that a quadrilateral is cyclic by using the angles resulting from its diagonals. Letβs begin by defining a cyclic quadrilateral. A cyclic quadrilateral is a four-sided polygon whose vertices are inscribed on a circle. For example, this is a cyclic quadrilateral. An inscribed angle is the angle made when two chords intersect on the circumference of the circle. The vertex of the angle lies on the circumference of the circle.

Before we consider the properties of a cyclic quadrilateral, letβs recall two very important theorems about inscribed angles. An angle π inscribed in a circle is half of the central angle two π that subtends the same arc on the circle. In other words, the angle at the circumference is half the angle at the center. This then leads into a second inscribed angle theorem, which tells us that inscribed angles subtended by the same arc are equal. So letβs see how these will be useful when we look at the proof of cyclic quadrilaterals.

We can take the example of this cyclic quadrilateral π΄π΅πΆπ·. And letβs draw in its diagonal line segments. Using the arc π·πΆ and given that the angles subtended by the same arc are equal, then we can say that the measure of angle π·π΄πΆ is equal to the measure of angle π·π΅πΆ. We can then use the same property with this arc π΄π΅ to show that the measure of angle π΄π·π΅ must be equal to the measure of angle π΄πΆπ΅.

We can then observe that, in any cyclic quadrilateral, the angle created by a diagonal and side is equal to the angle created by the other diagonal and opposite side. In this example, we found two pairs of congruent angles. But we can also use the arc π΅πΆ to show that the measure of angle π΅π΄πΆ is equal to the measure of angle π΅π·πΆ. Using the arc π΄π· would show that the measure of angle π΄π΅π· is equal to the measure of angle π΄πΆπ·.

When it comes to proving if a quadrilateral is cyclic, weβll need to see if the converse of this theorem is true. Letβs see if we can prove that if the angles created by the diagonals are equal, then that means that the quadrilateral is cyclic. Letβs take a different quadrilateral, π΄π΅πΆπ·, along with its diagonals. If we can prove that the measure of angle π·π΄πΆ is equal to the measure of angle π·π΅πΆ, then the quadrilateral is cyclic. This is because π·πΆ must be an arc of the circle. Therefore, π΄ and π΅ must also be points on the same circle. Therefore, every vertex must be on the circle. And this is by definition a cyclic quadrilateral.

Of course, it doesnβt always have to be the top two angles here that we proved that are congruent. For example, if we could prove that the measure of angle π΄π·π΅ is equal to the measure of angle π΄πΆπ΅, then this would also demonstrate that the quadrilateral is cyclic. However, we just need one of these pairs of congruent angles to demonstrate that the quadrilateral is cyclic. You might also wonder if perhaps every quadrilateral is cyclic. But letβs have a look at a different example.

Here is quadrilateral πΈπΉπΊπ». We can observe by eye that the measure of angle πΊπΈπ» is not equal to the measure of angle π»πΉπΊ. We could not draw a circle that passes through all four vertices. And so πΈπΉπΊπ» is not a cyclic quadrilateral.

Weβll now look at some examples where we prove if a quadrilateral is cyclic or not.

Is there a circle passing through the vertices of the quadrilateral π΄π΅πΆπ·?

If there is a circle passing through the vertices of this quadrilateral, then it would be a cyclic quadrilateral. There are a number of different angle properties we can use to determine if this quadrilateral is cyclic. However, given we have the diagonals marked, letβs check the angles made with the diagonals. We can then pose the question, is there an angle made with the diagonal and side which is equal in measure to the angle created by the other diagonal and opposite side?

At the minute, we donβt have any congruent pairs of angles in the diagram. However, we might observe that this angle πΆπ΄π΅ is an angle created by a side and diagonal. Angle πΆπ·π΅ is the angle created by the other diagonal and the opposite side. So letβs see if itβs congruent to angle πΆπ΄π΅. Letβs consider the triangle πΆπ΅π· and recall that the interior angles in a triangle sum to 180 degrees. Therefore, we can say that the three angles in this triangle, 54 degrees plus 79 degrees plus the measure of angle πΆπ·π΅, must be equal to 180 degrees. We can simplify the left-hand side and then subtract 133 degrees from both sides, which gives us that the measure of angle πΆπ·π΅ is 47 degrees.

This means that we now have a pair of congruent angles. The measure of angle πΆπ·π΅ is equal to the measure of angle πΆπ΄π΅. Therefore, we can say that an angle made with a diagonal and side is equal in measure to the angle created by the other diagonal and opposite side. Therefore, we can give the answer yes. Since weβve shown that this is a cyclic quadrilateral, we could draw a circle which passes through all four vertices of π΄π΅πΆπ·.

Letβs look at another example.

We can observe in this figure that we have the two diagonals marked, π΄πΆ and π΅π·. If we can prove that an angle created by a diagonal and side is equal in measure to the angle created by the other diagonal and opposite side, then the quadrilateral is cyclic. If they are not equal, then it is not a cyclic quadrilateral. The angle π΅π·π΄ is an angle created by a diagonal and side. The angle created by the other diagonal and opposite side is angle π΅πΆπ΄. If these two angles are congruent, then the quadrilateral is cyclic.

Another pair of angles we could check would be angle π·π΄πΆ and angle π·π΅πΆ. If we knew that just one pair of these angle measures were congruent, then that would be sufficient to show that the quadrilateral is cyclic. So letβs see if we can work out this angle measure of π΅πΆπ΄.

We are given that this angle measure π΅πΈπ΄ is 90 degrees. And we can remember that the angles on a straight line sum to 180 degrees. So that means that angle π΅πΈπΆ must also measure 90 degrees. We can now consider the triangle π΅πΈπΆ and remember that the angles in a triangle add up to 180 degrees. And so we can write that 63 degrees plus 90 degrees plus the measure of angle π΅πΆπΈ is 180 degrees. Therefore, the measure of angle π΅πΆπΈ is equal to 180 degrees subtract 153 degrees, which is 27 degrees. Therefore, these two angle measures created at the diagonals are not equal. And this means that π΄π΅πΆπ· is not a cyclic quadrilateral. Therefore, we can give the answer no.

At the start of this question, we did also say that we could check the angle measures of πΆπ΅π· and πΆπ΄π·. Calculating that angle π΄πΈπ· is also 90 degrees, we could then establish that angle πΆπ΄π· is 52 degrees. But of course 63 degrees is not equal to 52 degrees, once again showing that π΄π΅πΆπ· is not a cyclic quadrilateral.

In the next question, weβll check if a given trapezoid is a cyclic quadrilateral.

Is the trapezoid π΄π΅πΆπ· a cyclic quadrilateral?

Given that we have a trapezoid, we should have one pair of parallel sides. And theyβre marked here. π΅πΆ and π΄π· are parallel. As we have a transversal π΅π·, we can work out that the angle πΆπ΅π· is alternate to the angle π΄π·π΅. Itβs also 84 degrees. We can label the intersection of the diagonals as point πΈ. And then we can take a closer look at triangle π΅πΈπΆ.

We can work out the measure of this unknown angle π΅πΆπΈ by remembering that the interior angles in a triangle add up to 180 degrees. Therefore, 84 degrees plus 52 degrees plus the measure of angle π΅πΆπΈ is equal to 180 degrees. 84 degrees plus 52 degrees gives 136 degrees. Subtracting 136 degrees from both sides gives us that the measure of angle π΅πΆπΈ is 44 degrees. So how does this help us work out if π΄π΅πΆπ· is cyclic or not?

Well, letβs consider that we were given an angle created by a diagonal and side, angle π΄π·π΅. Weβve just calculated the angle created by the other diagonal and opposite side. If these two angle measures are congruent, then the quadrilateral is cyclic. But of course 84 degrees is not equal to 44 degrees. And so the two angle measures are not congruent. Therefore, π΄π΅πΆπ· is not cyclic. We can therefore answer the question with no.

So far, we have seen specific examples of different quadrilaterals. However, in the next two examples, weβll consider general statements about sets of quadrilaterals, beginning with determining if all rectangles are cyclic quadrilaterals or not.

All rectangles are cyclic quadrilaterals. (A) True or (B) false.

We can begin by recalling that a cyclic quadrilateral is a quadrilateral with all four vertices inscribed on a circle. One way we can prove that a quadrilateral is cyclic is by demonstrating that an angle created by a diagonal and side is equal in measure to the angle created by the other diagonal and opposite side. So letβs consider the properties of a rectangle, which is defined as a quadrilateral with four angles of 90 degrees. In a rectangle, there are also two pairs of opposite congruent sides.

We know that the diagonals of a rectangle divide the rectangle into two congruent triangles. Together, both diagonals create four congruent triangles. That is, any triangle thatβs created by two sides and a diagonal is congruent to any other triangle thatβs also created by two sides and a diagonal. So, for this rectangle, which we can call π½πΎπΏπ, we can say that triangle π½πΎπΏ is congruent to triangle πΏππ½. We can also say that this same triangle πΏππ½ is congruent to triangle ππΏπΎ.

Within this last pair of congruent triangles, we can also state that two corresponding angle measures are congruent since the measure of angle ππ½πΏ is equal to the measure of angle πΏπΎπ. This means that an angle created by a diagonal and side is equal in measure to the angle created by the other diagonal and opposite side. And so this rectangle and indeed any rectangle must be a cyclic quadrilateral.

As an alternative method, we could also consider that the diagonals of a rectangle are equal in length and bisect each other. This means that there will be four equal-length line segments extending from the point of intersection. These can be considered as radii extending from the center of a circle. Either method will allow us to give the answer true, since all rectangles are cyclic quadrilaterals.

In this example, we established that all rectangles are cyclic. However, itβs important to note that squares, which are defined as quadrilaterals with all sides equal and all internal angles equal to 90 degrees, are a subset of rectangles. Therefore, all squares are also cyclic quadrilaterals.

Weβll now look at a different type of quadrilateral.

All isosceles trapezoids are cyclic quadrilaterals. (A) True or (B) false.

Letβs begin by recalling that a trapezoid is a quadrilateral with one pair of parallel sides. An isosceles trapezoid is a special type of trapezoid which has the additional property that the two nonparallel sides, which are sometimes called legs, are equal in length. So letβs draw an isosceles trapezoid. It has one pair of parallel sides, and the other two nonparallel sides are equal in length. We can then use one of the properties of the diagonals of an isosceles trapezoid. The diagonals of an isosceles trapezoid create two congruent triangles at the legs. They also create two similar triangles at the bases.

Be careful, however, this rule only applies to isosceles trapezoids. For example, letβs take this trapezoid which is not isosceles. The diagonals do not create two congruent triangles at the legs. But letβs return to the isosceles trapezoid and look at the angles. We can define the intersection of the diagonals as point πΈ. And then because we have two congruent triangles, we can identify corresponding angles. The measure of angle π·π΄πΈ must be equal to the measure of angle πΆπ΅πΈ. Furthermore, the measure of angle π΄π·πΈ is equal to the measure of angle π΅πΆπΈ. Either of these pairs of congruent angles would demonstrate that the measure of an angle created by a diagonal and side is equal to the angle created by the other diagonal and opposite side. This means that isosceles trapezoid π΄π΅πΆπ·, and any isosceles trapezoid, is a cyclic quadrilateral.

Although not all trapezoids are cyclic quadrilaterals, all isosceles trapezoids are. So we can give the answer true.

We can summarize the previous two examples by saying that although there are some quadrilaterals which may be shown to be cyclic, there are three types which will always be cyclic. They are squares, rectangles, and isosceles trapezoids.

Now, letβs summarize the key points of this video. We began by noting that a cyclic quadrilateral is a four-sided polygon whose vertices are inscribed on a circle. In a cyclic quadrilateral, the angle created by a diagonal and side is equal in measure to the angle created by the other diagonal and opposite side. The converse of this is also true. Thus, one method we can use to prove a quadrilateral is cyclic is by demonstrating that an angle created by a diagonal and side is equal in measure to the angle created by the other diagonal and opposite side. Finally, we also saw that all squares, rectangles, and isosceles trapezoids are cyclic quadrilaterals.

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