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Question Video: Finding the Measures of Angles in a Quadrilateral given a Relation between Them by Solving Linear Equations Mathematics • 11th Grade

In the parallelogram ๐ธ๐น๐บ๐ป, ๐‘šโˆ ๐ธ๐ป๐บ = (3๐‘ฅ + 19)ยฐ, ๐‘šโˆ ๐ป๐บ๐น = (2๐‘ฅ + 6)ยฐ, and ๐‘šโˆ ๐น๐ธ๐ป = (2๐‘ฆ)ยฐ. Find the values of ๐‘ฅ and ๐‘ฆ.

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Video Transcript

In the parallelogram ๐ธ๐น๐บ๐ป, the measure of angle ๐ธ๐ป๐บ is three ๐‘ฅ plus 19 degrees, the measure of angle ๐ป๐บ๐น is two ๐‘ฅ plus six degrees, and the measure of angle ๐น๐ธ๐ป is two ๐‘ฆ degrees. Find the values of ๐‘ฅ and ๐‘ฆ.

Letโ€™s begin by adding the expressions that weโ€™re given for these three angles onto the diagram. In order to answer this question, weโ€™re going to need to recall some key facts about properties of angles in parallelograms.

There are two key properties that we need. Firstly, opposite angles in a parallelogram are congruent. They are the same as each other. Secondly, adjacent angles are supplementary, which means that they sum to 180 degrees.

Letโ€™s apply this second property to the angles ๐ธ๐ป๐บ and ๐ป๐บ๐น, which are a pair of adjacent angles. As these angles are supplementary, we can form an equation using the unknown variable ๐‘ฅ. Three ๐‘ฅ plus 19 plus two ๐‘ฅ plus six is equal to 180. The left-hand side of this equation simplifies to five ๐‘ฅ plus 25.

In order to find the value of ๐‘ฅ, we now need to solve this equation. The first step is to subtract 25 from both sides, giving five ๐‘ฅ is equal to 155. The next step is to divide both sides of the equation by five, giving ๐‘ฅ is equal to 31. So weโ€™ve found the value of ๐‘ฅ, and now we need to consider how to find the value of ๐‘ฆ.

There are two possible ways that you could do this. We could use the first property of angles in parallelograms, that opposite angles are congruent, and therefore angle ๐น๐ธ๐ป is equal to angle ๐ป๐บ๐น. This would give the equation two ๐‘ฆ is equal to two ๐‘ฅ plus six. And remember, we already know the value of ๐‘ฅ. Itโ€™s 31.

Therefore, we have that two ๐‘ฆ is equal to two multiplied by 31 plus six. This simplifies to give two ๐‘ฆ is equal to 68. And dividing both sides of the equation by two, we see that ๐‘ฆ is equal to 34. The other method that you could choose to use is to stick with the second fact, that adjacent angles are supplementary. This means that angle ๐น๐ธ๐ป plus angle ๐ธ๐ป๐บ is equal to 180.

You could therefore form an equation, two ๐‘ฆ plus three ๐‘ฅ plus 19 is equal to 180. By substituting ๐‘ฅ equals 31, you could then solve the equation to find the value of ๐‘ฆ. And of course, it would give the same result. You can confirm that yourself if you wish to. The solution to the problem is ๐‘ฅ is equal to 31, ๐‘ฆ is equal to 34.

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