# 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 πΈπΉπΊπ», πβ πΈπ»πΊ = (3π₯ + 19)Β°, πβ π»πΊπΉ = (2π₯ + 6)Β°, and πβ πΉπΈπ» = (2π¦)Β°. Find the values of π₯ and π¦.

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### Video Transcript

In the parallelogram πΈπΉπΊπ», the measure of angle πΈπ»πΊ is three π₯ plus 19 degrees, the measure of angle π»πΊπΉ is two π₯ plus six degrees, and the measure of angle πΉπΈπ» is two π¦ degrees. Find the values of π₯ and π¦.

Letβs begin by adding the expressions that weβre given for these three angles onto the diagram. In order to answer this question, weβ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.

Letβs apply this second property to the angles πΈπ»πΊ and π»πΊπΉ, which are a pair of adjacent angles. As these angles are supplementary, we can form an equation using the unknown variable π₯. Three π₯ plus 19 plus two π₯ plus six is equal to 180. The left-hand side of this equation simplifies to five π₯ plus 25.

In order to find the value of π₯, we now need to solve this equation. The first step is to subtract 25 from both sides, giving five π₯ is equal to 155. The next step is to divide both sides of the equation by five, giving π₯ is equal to 31. So weβve found the value of π₯, and now we need to consider how to find the value of π¦.

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 πΉπΈπ» is equal to angle π»πΊπΉ. This would give the equation two π¦ is equal to two π₯ plus six. And remember, we already know the value of π₯. Itβs 31.

Therefore, we have that two π¦ is equal to two multiplied by 31 plus six. This simplifies to give two π¦ is equal to 68. And dividing both sides of the equation by two, we see that π¦ 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 πΉπΈπ» plus angle πΈπ»πΊ is equal to 180.

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