In the given diagram, 𝑥 and 𝑦 are two adjacent angles that lie on the same line. Find an equation for their sum.
We’re given a diagram to help us to work out the answer to this problem. And the first piece of information that describes our diagram is the fact that 𝑥 and 𝑦 are two adjacent angles. We can tell that they’re adjacent angles because they share a common vertex, that’s this point here, and also a common side. Because of these two features, we can tell that these are two angles that are next door to each other. They’re adjacent.
The second piece of information that we’re told is that they lie on the same line. And we can see by looking at the diagram that because it’s the same line, this is a straight line. Now, what if there was only one angle along a straight line? What do we know about straight-line angles? Straight-line angles are the same as half a turn. And these have a value of 180 degrees. We can use this fact to help us solve the problem.
We know that the sum of two angles is the total of those angles. It’s what we get when we add them together. And we’re asked to find an equation for the sum of angles 𝑥 and 𝑦. Well, if the word sum means total, then our equation is going to have to be 𝑥 plus 𝑦 equals something. What are we going to get if we add angles 𝑥 and 𝑦 together? As we’ve said already, 𝑥 and 𝑦 together make a straight line. So the total of 𝑥 and 𝑦 is going to equal the number of degrees in a straight-line angle. And we know that this is 180 degrees.
We don’t know what angles 𝑥 and 𝑦 are. But what we do know is that when we add them together, we get 180 degrees. 𝑥 could be 80 degrees. And 𝑦 could be 100. 𝑥 could be 75. And 𝑦 could be 105. We don’t know. But we’ve used what we do know to find an equation for their total. And we can say that 𝑥 plus 𝑦 equals 180 degrees, a straight-line angle.