Question Video: Deciding If Polygons Are Convex or Concave Mathematics

Video Transcript

In a pentagon 𝐴𝐵𝐶𝐷𝐸, the measure of angle 𝐴 is 110 degrees. The measure of angle 𝐷 is equal to 62 degrees. The measure of angle 𝐸 is equal to 89 degrees. And the measure of angle 𝐵 is equal to the measure of angle 𝐶. Decide whether 𝐴𝐵𝐶𝐷𝐸 is convex or concave.

So in this question, what we need to do is decide whether 𝐴𝐵𝐶𝐷𝐸 is convex or concave. So we need to know whether a polygon is going to be, like we said, convex or concave. So a polygon is convex if all the interior angles are less than 180 degrees. But otherwise, it is concave. So if we’ve got any of the angles greater than 180 degrees, then it’ll be concave.

Well, in this question, we know angle 𝐴, 𝐷, and 𝐸. But we don’t know angles 𝐵 and 𝐶. So what we need to do now is work them out. So first of all, we need to know the sum of the interior angles of a pentagon. We know that’s 540 degrees. One reason we know that is — well, I’ve drawn a sketch here to show it.

We have our five-sided shape, so our pentagon. And we split it in to triangles, starting at one vertex and going down to the other vertices that will make triangles. And we can see here that we can make three triangles. And we know that the interior angles of our triangle add up to 180 degrees. So therefore, we’ve got three of those, which gonna give us our 540 degrees.

There’s also a formula we could’ve used, which is 180 multiplied by 𝑛 minus two, where 𝑛 is the number of sides, which would’ve given us 180 multiplied by five minus two, which once again would’ve given us 180 multiplied by three, which would be our 540.

Okay great, so we know the sum of the interior angles of our pentagon. What next? Well, we know that angle 𝐴, 𝐷, and 𝐸 are all less than 180 degrees. But we need to work out angle 𝐵 and angle 𝐶 like we stated in the beginning.

So as we know that the measure of angle 𝐵 is equal to the measure of angle 𝐶, we can say that the measure of angle 𝐵 plus the measure of angle 𝐶 is gonna be equal to 540, which is the sum of the interior angles of our pentagon minus then all the other angles added together, so 110 plus 62 plus 89. So therefore, we can say that the measure of angle 𝐵 plus the measure of angle 𝐶 is gonna be equal to 279 degrees.

And as we’ve already stated, we know that the measure of angle 𝐵 is equal to the measure of angle 𝐶. So therefore, if they’re equal to each other, what we can do is divide 279 degrees by two to find one of them. So when we do that, we get 139.5 degrees. So therefore, we know that the measure of angle 𝐵 is equal to 139.5 degrees. And the measure of angle 𝐶 is equal to 139.5 degrees.

So therefore, we have now calculated all the angles in our pentagon. And we know that they’re all less than 180 degrees. So we can say that our pentagon is convex.