Video: Finding the Measures of Two Vertically Opposite Angles by Solving Two Linear Equations

Two intersecting straight lines are shown. Find the values of π‘₯ and 𝑦.

04:34

Video Transcript

Two intersecting straight lines are shown. Find the values of π‘₯ and 𝑦.

As the question says, we have a pair of intersecting straight lines. And when this is the case, we need to look out for vertically opposite angles. Well, we have two pairs of those. These two angles are vertically opposite from one another. And the angle labeled as 𝑦 is vertically opposite to this one. In fact, though, we’re just going to begin by worrying about π‘₯.

Next, we state the following fact. Vertically opposite angles are equal. Now, we said that the angle nine π‘₯ minus 30 degrees is vertically opposite to the angle seven π‘₯ minus four degrees. This must mean then that we can say that nine π‘₯ minus 30 is equal to seven π‘₯ minus four. Our job now is to solve this equation for π‘₯.

Now, since we have π‘₯s on both sides of our equation, our first job is to get rid of the smallest number of π‘₯. Seven π‘₯ is less than nine π‘₯, so we’re going to subtract seven π‘₯ from both sides of our equation. Nine π‘₯ minus seven π‘₯ is two π‘₯. So, the left-hand side of our equation is two π‘₯ minus 30. On the right-hand side, we’re simply left with negative four.

Next, we want to get rid of the negative 30. So, we do the inverse operation; we add 30 to both sides. On the left-hand side, that leaves us with two π‘₯. And negative four plus 30 is positive 26. So, our equation is two π‘₯ equals 26. Remembering, of course, that two π‘₯ means two times π‘₯, we know now that we need to divide both sides of the equation by two. Two π‘₯ divided by two is π‘₯. And 26 divided by two is 13.

So, we’ve calculated π‘₯ to be equal to 13. We’re not quite finished though. The question also asks us to find the value of 𝑦. So, what do we do next? Well, there are a couple of things we could do. One of the facts we could use is that angles on a straight line sum to 180 degrees. We could also use the fact that angles about a point sum to 360 degrees. But, here, that requires a little more work. So, we’re going to calculate the value of either seven π‘₯ minus four or nine π‘₯ minus 30.

Remember, that will give us the same value. Let’s calculate seven π‘₯ minus four when π‘₯ is equal to 13. It’s seven times 13 minus four, which is 87. So, this angle here is 87 degrees. We can set up and solve an equation for 𝑦. We know that angles on a straight line sum to 180 degrees, so we say that 87 plus 𝑦 equals 180. This time, we solve by subtracting 87 from both sides. And we see that 𝑦 is equal to 93. π‘₯ is equal to 13, and 𝑦 is equal to 93.

Now, in fact, with angle problems, there’s often more than one way to solve that same problem. In this case, we could have originally started by quoting the fact that angles on a straight line sum to 180 degrees. We form one equation by adding these two angles. And we know that equals 180. So, we get nine π‘₯ minus 30 plus 𝑦 equals 180. Now, let’s combine the numerical parts by adding 30 to both sides. And when we do, we find that nine π‘₯ plus 𝑦 equals 210.

Next, we use the same idea, this time adding these two angles. We get seven π‘₯ minus four plus 𝑦 equals 180. To combine the numerical parts, we add four to both sides. And we find that seven π‘₯ plus 𝑦 equals 184. Notice, we now have a pair of simultaneous equations. And we can see that the coefficient, the number of 𝑦s we have in each equation, is the same. Since the signs of the coefficient of 𝑦 is also the same, we subtract both equations.

Nine π‘₯ minus seven π‘₯ is two π‘₯. 𝑦 minus 𝑦 is zero. And 210 minus 184 is 26. To solve for π‘₯, we divide through by two. And we once again find π‘₯ to be equal to 13 degrees. We find the value of 𝑦 by substituting π‘₯ into either of our original equations. If we substitute it into the second one, we get seven times 13 plus 𝑦 equals 184. Seven times 13 is 91. And then, we solve for 𝑦 by subtracting 91 from both sides. And once again, we find 𝑦 to be equal to 93.

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