Question Video: Finding the Measures of Angles in a Quadrilateral given a Relation between Them by Solving Linear Equations | Nagwa Question Video: Finding the Measures of Angles in a Quadrilateral given a Relation between Them by Solving Linear Equations | Nagwa

# Question Video: Finding the Measures of Angles in a Quadrilateral given a Relation between Them by Solving Linear Equations Mathematics

From the figure, in which πβ πΆπ·π΄ = 5π₯Β°, πβ π΅πΆπ· = 7π¦Β°, and πβ π΄π΅πΆ = 8π¦Β°, find the values of π₯ and π¦.

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### Video Transcript

From the figure, in which the measure of angle πΆπ·π΄ is five π₯ degrees, the measure of angle π΅πΆπ· is seven π¦ degrees, and the measure of angle π΄π΅πΆ is eight π¦ degrees, find the values of π₯ and π¦.

Letβs begin by adding what weβve been told to our figure. The measure of angle πΆπ·π΄ is five π₯; thatβs this one. The angle π΅πΆπ· is seven π¦; thatβs this one. And the measure of angle π΄π΅πΆ, thatβs this one, is eight π¦. Okay, so how does this help us? Well, we have a couple of useful facts. Firstly, we know that the sum of the interior angles in a quadrilateral, thatβs a four-sided polygon, is 360 degrees. So according to our diagram, we can say that eight π¦ plus seven π¦ plus five π₯ plus 85 equals 360. Letβs add seven π¦ and eight π¦ to get 15π¦.

Then we spot that we have two parallel lines in our diagram. The parallel lines are π΅π΄ and πΆπ·. We can quote that cointerior angles sum to 180 degrees. So these two angles, 85 and 5π₯, are cointerior. They add to 180. And we form a second equation; this time, 85 plus five π₯ equals 180. Letβs solve this equation for π₯. We subtract 85 from both sides, so five π₯ is equal to 95. Then we divide through by five. So π₯ is 95 divided by five, which is 19. Remember, we still need to calculate the value of π¦. So letβs clear some space.

Weβre going to go back to the first equation we formed and substitute π₯ equals 19. When we do, we get 15π¦ plus five times 19 plus 85 equals 360. Thatβs 15π¦ plus 95 plus 85 equals 360. 95 plus 85 is 180. So we have an equation in π¦ that we can solve. We subtract 180 from both sides. So 15π¦ is 180. And then, we divide through by 15. π¦ is 180 divided by 15, which is equal to 12. So π₯ is 19 and π¦ is equal to 12.

Now, with angle questions, thereβs often more than one way to answer them. We could have used the fact that the angles at π΅ and πΆ are cointerior. They sum to 180 degrees. We form an equation, and we get eight π¦ plus seven π¦ equals 180. So 15π¦ is equal to 180. Once again, we solve this equation by dividing by 15. And we get π¦ is equal to 12.

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