Question Video: Using the Properties of Cyclic Quadrilaterals to Solve Problems | Nagwa Question Video: Using the Properties of Cyclic Quadrilaterals to Solve Problems | Nagwa

# Question Video: Using the Properties of Cyclic Quadrilaterals to Solve Problems Mathematics • Third Year of Preparatory School

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In the figure, πβ π΄π·πΈ = 26.9Β°, πβ π΄πΈπ΅ = 82.9Β°, and πβ π΄π΅πΈ = 59.1Β°. Is π΄π΅πΆπ· a cyclic quadrilateral?

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### Video Transcript

In the figure below, the measure of angle π΄π·πΈ equals 26.9 degrees, the measure of angle π΄πΈπ΅ equals 82.9 degrees, and the measure of angle π΄π΅πΈ equals 59.1 degrees. Is π΄π΅πΆπ· a cyclic quadrilateral?

We can recall that a cyclic quadrilateral is a quadrilateral that has all four vertices inscribed on a circle. One angle property we can use to prove a quadrilateral is cyclic is that if 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.

So letβs have a look at this quadrilateral. And weβll begin by filling in the angle measures we were given. Here are the angle measures then. π΄π·πΈ is 26.9 degrees, π΄πΈπ΅ is 82.9 degrees, and π΄π΅πΈ is 59.1 degrees. We might notice then that within this triangle of π΄π΅πΈ, we have two angle measures and one unknown. Using the fact that the interior angles in a triangle add up to 180 degrees will allow us to calculate the measure of angle πΈπ΄π΅. So weβll have 82.9 degrees plus 59.1 degrees plus the measure of angle πΈπ΄π΅ is equal to 180 degrees.

Simplifying 82.9 degrees plus 59.1 degrees gives us 142 degrees. Subtracting 142 degrees from both sides leaves us with the measure of angle πΈπ΄π΅ equal to 38 degrees. At this point, however, we still donβt have the measures of a pair of angles made at the diagonals. For example, if we consider this angle at πΈπ΄π΅ and see if itβs equal to the angle made by the other diagonal and opposite side, that is, angle πΆπ·π΅, then we could check if the quadrilateral is cyclic. An alternative pair of angles we could check would be angles π·πΆπ΄ and angle π·π΅π΄.

For either angle pair, weβll need to keep going and see what other angles we can work out in this figure. The fact that π·π΅ is a straight line will allow us to calculate this angle measure of π΄πΈπ·. So the measure of angle π΄πΈπ· is equal to 180 degrees subtract 82.9 degrees, which leaves us with 97.1 degrees. Observing then that we have this triangle π΄πΈπ· and two known angles will allow us to calculate the third angle π·π΄πΈ. Because these three angles will add to 180 degrees, we can write that 26.9 degrees plus 97.1 degrees plus the measure of angle π·π΄πΈ is equal to 180 degrees.

We can simplify this and then subtract 124 degrees from both sides of the equation. And so the measure of angle π·π΄πΈ is 56 degrees.

In order to do any further calculations in this figure, weβll need to observe some important markings. We are given that the line segments π΄π· and πΆπ· are congruent. When we consider these line segments as part of the triangle π΄πΆπ·, then that means that triangle π΄πΆπ· is an isosceles triangle. An isosceles triangle has two equal sides and two equal angle measures. And that means that the measure of angle π·π΄πΆ must be equal to the measure of angle π·πΆπ΄. And so theyβll both be 56 degrees.

So remember that weβve done all these angle measurement calculations in order to see if we have a pair of equal angles at the diagonals. And we do now have a pair of angles which we can compare: angle π΄π΅π· and angle π΄πΆπ·. And we can see that of course 56 degrees is not equal to 59.1 degrees. This means that an angle created by a diagonal and side is not equal to the angle created by the other diagonal and opposite side. And so we can give the answer no, because we have proved that π΄π΅πΆπ· is not a cyclic quadrilateral.

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