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Video: Finding the Measure of the Angles in a Quadrilateral given a Relation between Them by Solving Linear Equations

Tim Burnham

𝐴𝐵𝐶𝐷 is a quadrilateral where 𝑚∠𝐴 = 165°, 𝑚∠𝐴𝐵𝐶 = 4𝑥°, 𝑚∠𝐵𝐶𝐷 = 4𝑥°, and 𝑚∠𝐶𝐷𝐴 = 5𝑥°. What is 𝑥?

04:53

Video Transcript

𝐴𝐵𝐶𝐷 is a quadrilateral where the measure of angle 𝐴 is one hundred and sixty-five degrees, the measure of angle 𝐴𝐵𝐶 is four 𝑥 degrees, the measure of angle 𝐵𝐶𝐷 is also four 𝑥 degrees, and the measure of angle 𝐶𝐷𝐴 is five 𝑥 degrees. What is 𝑥?

Well we’ve been given a diagram of the quadrilateral, so we can put the measurements that we were given in the question onto the diagram. The measure of angle 𝐴 is one hundred and sixty-five degrees, so that’s that one there; the measure of angle 𝐴𝐵𝐶, so that’s this one here, is four 𝑥 degrees; the measure of angle 𝐵𝐶𝐷, that’s this one here, is also four 𝑥 degrees; and the measure of angle 𝐶𝐷𝐴, so that’s this one here, is five 𝑥 degrees.

So we need to work out the value of 𝑥. Well the first piece of information we need to know is what is the sum of the interior angles in a quadrilateral. Well if we take a quadrilateral and we pick a vertex, we can split the quadrilateral into two triangles.

Now we know that the sum of the measures of the angles in a triangle is a hundred and eighty degrees, so we know that this angle here plus this angle here plus this angle here is a hundred and eighty degrees. And we also know that this angle here plus this angle here plus this angle here also is a hundred and eighty degrees.

Now in doing that, we’ve counted each part of each angle in our quadrilateral just once. And if the three red angles add up to a hundred and eighty and the three green angles add up to a hundred and eighty, then the four pink angles must add up to a hundred and eighty plus a hundred and eighty. So, the sum of the measures of the interior angles in a quadrilateral is three hundred and sixty degrees.

Now that means that if I add together the measures of all these angles here, I will also get a total of three hundred and sixty degrees. So that means that a hundred and sixty-five plus four 𝑥 plus four 𝑥 plus five 𝑥 is three hundred and sixty. Well I’ve got three terms involving 𝑥. I’ve got four 𝑥𝑠 plus another four 𝑥𝑠 plus five more 𝑥𝑠, so that gives me eight 𝑥𝑠 plus five 𝑥𝑠, which is thirteen 𝑥𝑠.

So a hundred and sixty-five plus thirteen 𝑥 is equal to three hundred and sixty. Now I want to know what 𝑥 is. So, I’ve got to try to get rid of this hundred and sixty-five. Now to do that, I can take away a hundred and sixty-five from the left-hand side. But to keep it as an equation, I also need to take away a hundred and sixty-five from the right-hand side, otherwise they won’t balance anymore. If I do that on the left, a hundred and sixty-five take away a hundred and sixty-five is nothing, so that just leaves me with thirteen 𝑥. And three hundred and sixty minus a hundred and sixty-five is one hundred and ninety-five.

So thirteen 𝑥 is equal to a hundred and ninety-five. So thirteen 𝑥 is equal to a hundred and ninety-five, but I want to know what one 𝑥 is equal to. Well thirteen 𝑥 means thirteen times 𝑥. So if I do the inverse operation of times-ing by thirteen, that’s dividing by thirteen, that will give me one 𝑥.

But of course if I divide the left-hand side by thirteen, then I’ve got to divide the right-hand side by thirteen as well, otherwise they won’t balance. So thirteen 𝑥 divided by thirteen is just 𝑥, and a hundred and ninety-five divided by thirteen is fifteen. So the answer is 𝑥 is equal to fifteen.

Now it’s always a good idea to check our answers, so then let’s go back and do that. This angle here, four 𝑥, four times fifteen, well two times fifteen is thirty and two times thirty is sixty, so this is sixty degrees. This is also sixty degrees. And five times fifteen is seventy-five, so this is seventy-five degrees.

So the sum of those interior angles of our quadrilateral there is a hundred and sixty-five plus sixty plus sixty plus seventy-five. Well, a hundred and sixty-five plus seventy-five is two hundred and forty, and sixty and sixty is a hundred and twenty. So our total is two hundred and forty plus a hundred and twenty is three hundred and sixty degrees. Well, that sounds right. That is the proper sum of the measures of the interior angles in a quadrilateral.

So, it looks like we’ve got a feasible and hopefully a correct answer.