# Question Video: Using the Properties of Cyclic Quadrilaterals to Solve Problems Mathematics • 11th Grade

Find πβ πΈππ, given that πΏπππΈ is a cyclic quadrilateral with πβ ππΏπΈ = 64Β° and πβ ππΈπ = 38Β°.

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### Video Transcript

Find the measure of angle πΈππ, given that πΏπππΈ is a cyclic quadrilateral with the measure of angle ππΏπΈ equal to 64 degrees and the measure of angle ππΈπ equal to 38 degrees.

In the diagram, we can see that the measure of angle ππΈπ is given as 38 degrees. So letβs also add in the fact that the measure of angle ππΏπΈ is 64 degrees. Here, weβre given that this quadrilateral is a cyclic quadrilateral. And that means that all four of the vertices would lie on a circle.

One important property of cyclic quadrilaterals which might be useful here is that opposite angles are supplementary. In other words, they add up to 180 degrees. The angle that weβre asked to find here is the measure of angle πΈππ, which is here on the quadrilateral. We might notice that this is also part of the triangle πΈππ. And if we knew the measure of angle πΈππ, then we might be able to find our unknown angle.

Angle πΈπΏπ is opposite to angle πΈππ. And so we know that these two angles will be supplementary. We can write the equation that these two angle measures must add to give 180 degrees. We were given that πΈπΏπ or indeed angle ππΏπΈ is 64 degrees. And so we can subtract that from both sides of the equation. And so the measure of angle πΈππ must be 116 degrees.

Now, we can use the two angles in triangle πΈππ and the fact that the interior angles in a triangle sum to 180 degrees to find the unknown angle. Therefore, we have 38 degrees plus 116 degrees plus the measure of angle πΈππ is equal to 180 degrees. We can then simplify this and subtract 154 degrees from both sides of the equation, which gives us the answer that the measure of angle πΈππ is 26 degrees.