In the following figure, 𝑧 equals two 𝑥 minus 69 and 𝑤 equals two 𝑦 minus 59. Find 𝑥 and 𝑦.
So we’ve been given a diagram in which one of the angles is known. It’s 83 degrees. And then we have angles of 𝑥, 𝑤, and 𝑧. And we’re also given information about 𝑧 and 𝑤 in terms of the variables 𝑥 and 𝑦. Let’s look carefully at the diagram in order to plan our approach to answering this question.
We can see that we have a pair of parallel lines indicated by the blue arrows that I’ve marked on these lines. We also have a transversal, a line cutting through these parallel lines. This may look more familiar if I extend each of the parallel lines and the transversal slightly.
Now what this tells me is that actually I can calculate the value of 𝑥 directly because I have a particular type of angles in parallel lines. The angle of 83 degrees and 𝑥 are what’s known as cointerior angles cause they both sit inside the parallel lines on the same side of the transversal.
So in order to calculate 𝑥, I need to remember key facts about cointerior angles. And it’s that they’re supplementary to one another, which means that the sum of the two angles is 180 degrees. So this means I can work out the value of 𝑥 by subtracting the other cointerior angle, 83 degrees, from 180. So we have that 𝑥 is equal to 97 degrees.
Now let’s look at the rest of the question. We’re asked to find 𝑥, which we’ve done, and 𝑦. Now we’re told about 𝑧 and 𝑤. And if I look carefully, 𝑧 is equal to two 𝑥 minus 69. So we have 𝑧 in terms of 𝑥. We also have 𝑤 is equal to two 𝑦 minus 59. So we have 𝑤 in terms of 𝑦.
Looking carefully at the diagram, we can also see that 𝑤 and 𝑧 are on a straight line, which means that a relationship exists between these two angles because angles on a straight line sum to 180 degrees. Thinking about all of this together suggests a possible approach that I can take.
As I know the value of 𝑥, I can calculate the value of 𝑧. It’s two 𝑥 minus 69. I can then use the fact that 𝑤 and 𝑧 are on a straight line together in order to calculate the value of 𝑤. Finally, once I know 𝑤, I’ll be able to solve an equation saying as 𝑤 is equal to two 𝑦 minus 59 in order to calculate 𝑦.
So this is the sequence of steps that we’re going to look to take in order to answer this problem. First of all, we’re looking to calculate 𝑧. Now 𝑧 remember is equal to two 𝑥 minus 69. So as we already know that 𝑥 is 97, 𝑧 must be two times 97 minus 69. So the value of 𝑧 is 125.
Next, we’re going to calculate 𝑤. And remember the key fact here was that angles on a straight line sum to 180 degrees. So in order to calculate 𝑤, we need to subtract the value of 𝑧 from 180. So we have that 𝑤 is equal to 180 minus 𝑧 or 180 minus 125. And this tells us that 𝑤 is equal to 55.
The final step then now in our 𝑤 is to calculate the value of 𝑦, which is the letter we’re originally asked to calculate. So remember that 𝑤 is equal to two 𝑦 minus 59. We now have that two 𝑦 minus 59 is equal to 55.
And here is an equation we can solve to find 𝑦. The first step is to add 59 to both sides of the equation. So we have that two 𝑦 is equal to 114. Next, we need to divide both sides of the equation by two. And in doing so, we have that 𝑦 is equal to 57. So finally, combining the two parts in order to give our answer to the question, we have that 𝑥 is equal to 97 and 𝑦 is equal to 57.