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

Video Transcript

Find the measure of angle 𝐸𝑀𝑁, given that 𝐿𝑀𝑁𝐸 is a cyclic quadrilateral with the measure of angle 𝑀𝐿𝐸 equal to 64 degrees and the measure of angle 𝑀𝐸𝑁 equal to 38 degrees.

In the diagram, we can see that the measure of angle 𝑀𝐸𝑁 is given as 38 degrees. So let’s also add in the fact that the measure of angle 𝑀𝐿𝐸 is 64 degrees. Here, we’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 we’re asked to find here is the measure of angle 𝐸𝑀𝑁, which is here on the quadrilateral. We might notice that this is also part of the triangle 𝐸𝑀𝑁. And if we knew the measure of angle 𝐸𝑁𝑀, then we might be able to find our unknown angle.

Angle 𝐸𝐿𝑀 is opposite to angle 𝐸𝑁𝑀. 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 𝐸𝐿𝑀 or indeed angle 𝑀𝐿𝐸 is 64 degrees. And so we can subtract that from both sides of the equation. And so the measure of angle 𝐸𝑁𝑀 must be 116 degrees.

Now, we can use the two angles in triangle 𝐸𝑀𝑁 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 𝐸𝑀𝑁 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 𝐸𝑀𝑁 is 26 degrees.