In the parallelogram 𝐸𝐹𝐺𝐻, the measure of angle 𝐸𝐻𝐺 is three 𝑥 plus 19 degrees, the measure of angle 𝐻𝐺𝐹 is two 𝑥 plus six degrees, and the measure of angle 𝐹𝐸𝐻 is two 𝑦 degrees. Find the values of 𝑥 and 𝑦.
Let’s begin by adding the expressions that we’re given for these three angles onto the diagram. In order to answer this question, we’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.
Let’s apply this second property to the angles 𝐸𝐻𝐺 and 𝐻𝐺𝐹, which are a pair of adjacent angles. As these angles are supplementary, we can form an equation using the unknown variable 𝑥. Three 𝑥 plus 19 plus two 𝑥 plus six is equal to 180. The left-hand side of this equation simplifies to five 𝑥 plus 25.
In order to find the value of 𝑥, we now need to solve this equation. The first step is to subtract 25 from both sides, giving five 𝑥 is equal to 155. The next step is to divide both sides of the equation by five, giving 𝑥 is equal to 31. So we’ve found the value of 𝑥, and now we need to consider how to find the value of 𝑦.
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 𝐹𝐸𝐻 is equal to angle 𝐻𝐺𝐹. This would give the equation two 𝑦 is equal to two 𝑥 plus six. And remember, we already know the value of 𝑥. It’s 31.
Therefore, we have that two 𝑦 is equal to two multiplied by 31 plus six. This simplifies to give two 𝑦 is equal to 68. And dividing both sides of the equation by two, we see that 𝑦 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 𝐹𝐸𝐻 plus angle 𝐸𝐻𝐺 is equal to 180.
You could therefore form an equation, two 𝑦 plus three 𝑥 plus 19 is equal to 180. By substituting 𝑥 equals 31, you could then solve the equation to find the value of 𝑦. And of course, it would give the same result. You can confirm that yourself if you wish to. The solution to the problem is 𝑥 is equal to 31, 𝑦 is equal to 34.