Lesson Video: Angles: Vertically Opposite Mathematics • 7th Grade

Video Transcript

In this video, we’ll learn how to identify vertically opposite angles and solve their problems. We’ll look at a short proof of the vertical angle theorem before considering a number of simple examples and examples that involve the use of algebra.

When two lines meet at a point in a plane, we say that they intersect; they’re intersecting lines. Now, if this is not the case, if two lines do not intersect, we say that they are parallel; they’ll never meet. For example, in this diagram, we see that lines 𝐴𝐵 and 𝐶𝐷 intersect at the point 𝑚.

So, what do we know about the angles here?

The two angles 𝐴𝑚𝐷 and 𝐵𝑚𝐶 are called what angles. Here, we look to introduce some terminology. We say that given a pair of intersecting lines, the angles which are opposite to one another form a pair of vertically opposite angles. That means angle 𝐴𝑚𝐷, which is this one, and 𝐵𝑚𝐶, which is this one, are vertically opposite to one another. Similarly, angle 𝐴𝑚𝐶, that’s this one, and 𝐵𝑚𝐷, which is this one, are also vertically opposite to one another. So, the phrase that goes in the blank here is “vertically opposite.”

So, what do we know about vertically opposite angles?

If two angles are vertically opposite, are they equal in measure? Consider two lines 𝐴𝐵 and 𝐶𝐷 which intersect one another at the point 𝑚. We now bring in an axiom. That’s just a statement that’s used in proof, which we take to be true. The axiom is that angles on a straight line sum to 180 degrees.

So, let’s take line 𝐴𝐵. We can say that angle 𝐴𝑚𝐷 and angle 𝐵𝑚𝐷 add to make 180 degrees. Similarly, we can say that angle 𝐴𝑚𝐶 and angle 𝐵𝑚𝐶 also add to make 180 degrees. And, in fact, if we consider the line 𝐶𝐷, we can form two further statements. That is, angle 𝐴𝑚𝐶 and 𝐴𝑚𝐷 add to make 180 as do angles 𝐵𝑚𝐶 and 𝐵𝑚𝐷.

So, we’re going to define one of our angles. Let’s define 𝐴𝑚𝐷 to be equal to 𝑥 degrees. And so, we can say in these two statements that 𝑥 plus 𝐵𝑚𝐷 equals 180 degrees, but also 𝐴𝑚𝐶 plus 𝑥 equals 180. We rearrange both of our equations by subtracting 𝑥 from both sides. Our first equation becomes 𝐵𝑚𝐷 equals 180 minus 𝑥. And our other equation becomes 𝐴𝑚𝐶 equals 180 minus 𝑥.

Now, notice, we’ve shown that both 𝐵𝑚𝐷 is equal to 180 minus 𝑥 and 𝐴𝑚𝐶 is equal to 180 minus 𝑥. These three dots mean “therefore,” and we can say that, therefore, 𝐵𝑚𝐷 must be equal to 𝐴𝑚𝐶. On our diagram, that’s this one and this one. Now, in fact, since this is the case and angles on a straight line sum to 180 degrees, we can show that both 𝐴𝑚𝐷 and 𝐵𝑚𝐶 are also equal. And so, we say that vertically opposite angles are equal. And so, the answer to this question is yes.

So, let’s have a look at how this can help us to solve problems.

Find the value of 𝑥 in the given diagram.

We say that this pair of angles here are vertically opposite to one another; they’re vertically opposite angles. And we can state the following. That is, vertically opposite angles are equal. And so, that must mean that this pair of angles, the pair of angles in our diagram, must be equal to one another. In other words, 𝑥 must be equal to 62. And that’s it. That’s the solution to this question. 𝑥 is equal to 62.

In our next example, we’ll look to increase the complexity a little to involve more than two angles.

Find the value of 𝑥.

Sometimes, it can be hard to spot what we need to do to find missing angles. So, instead, we just consider what we do actually know and go from there. In fact, there’s usually more than one method, as in this question. So, let’s consider both. Let’s begin by looking at two lines. We have the straight line 𝐴𝐶 and the straight line 𝐵𝐷. They intersect at a point; they’re intersecting lines. And so, we’re going to find some pairs of vertically opposite angles.

Now, if we look carefully, we see that this angle here is vertically opposite to this angle here. Now, we know that vertically opposite angles are equal. And so, the sum of the two angles between 𝐴 and 𝐷, that’s 𝑥 and 74, must be equal to 144. We can form an equation, that is, 144 equals 𝑥 plus 74. We solve this equation for 𝑥 by subtracting 74 from both sides. So that we get 𝑥 being equal to 144 minus 74. Well, 144 minus 74 is 70. So, we see that 𝑥 is equal to 70. And that angle is 70 degrees.

But what else could we have done? Well, we know that angles on a straight line add to make 180 degrees. This means this angle and this angle must sum to 180. 180 minus 144 is 36. So, we have an angle of 36 degrees here. Next, we use the same fact; that is, angles on a straight line sum to 180. And we know that this angle, this angle, and this angle must make 180. We can, therefore, form an equation. We can say that 𝑥 plus 74 plus 36 equals 180. 74 plus 36 is 110. So, this becomes 𝑥 plus 110 equals 180. And we solve for 𝑥 by subtracting 110 from both sides. Giving us, once again, 𝑥 is equal to 70.

In our next example, we’ll consider how two-step algebraic equations can help us solve these problems.

In the following figure, find the value of 𝑥.

In the diagram, we have two intersecting lines. These two angles here, we say, are vertically opposite angles. And so, we recall what we know about vertically opposite angles. We know that they’re equal. We say vertically opposite angles are equal. And so, this means our two angles, two 𝑥 plus five and 67, must be equal to one another. We write this as an equation: two 𝑥 plus five equals 67.

And we can solve this equation to find the value of 𝑥 by performing a series of inverse operations. We begin by subtracting five from both sides. When we subtract five from the left-hand side, we’re left with two 𝑥. And 67 minus five is 62. So, our equation is now two 𝑥 equals 62. Currently, the 𝑥 is being multiplied by two. And the inverse operation here then is to divide through by two. Two 𝑥 divided by two is 𝑥. And 62 divided by two is 31. So, we’ve calculated 𝑥 to be equal to 31.

Now, since we’ve worked with algebra, it’s sensible to check our answer by substituting it back into the original expression. Two 𝑥 means two times 𝑥. And we found 𝑥 to be 31. So, we multiply 31 by two and add five. That’s 62 plus five, which is equal to 67. We know that this must be equal to 67 since vertically opposite angles are equal. So, that’s an indication to us that we’ve performed our calculations correctly. 𝑥 is equal to 31.

In our final example, we’ll consider how to solve such problems when we have two algebraic expressions.

Two intersecting straight lines are shown. Find the values of 𝑥 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 𝑥.

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

Now, since we have 𝑥s on both sides of our equation, our first job is to get rid of the smallest number of 𝑥. Seven 𝑥 is less than nine 𝑥, so we’re going to subtract seven 𝑥 from both sides of our equation. Nine 𝑥 minus seven 𝑥 is two 𝑥. So, the left-hand side of our equation is two 𝑥 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 𝑥. And negative four plus 30 is positive 26. So, our equation is two 𝑥 equals 26. Remembering, of course, that two 𝑥 means two times 𝑥, we know now that we need to divide both sides of the equation by two. Two 𝑥 divided by two is 𝑥. And 26 divided by two is 13.

So, we’ve calculated 𝑥 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 𝑥 minus four or nine 𝑥 minus 30.

Remember, that will give us the same value. Let’s calculate seven 𝑥 minus four when 𝑥 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. 𝑥 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 𝑥 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 𝑥 plus 𝑦 equals 210.

Next, we use the same idea, this time adding these two angles. We get seven 𝑥 minus four plus 𝑦 equals 180. To combine the numerical parts, we add four to both sides. And we find that seven 𝑥 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 𝑥 minus seven 𝑥 is two 𝑥. 𝑦 minus 𝑦 is zero. And 210 minus 184 is 26. To solve for 𝑥, we divide through by two. And we once again find 𝑥 to be equal to 13 degrees. We find the value of 𝑦 by substituting 𝑥 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.

In this video, we saw that when two lines meet at a point in a plane, they intersect; we call them intersecting lines, as shown. We also say that in a pair of intersecting lines, the angles which are opposite to each other form a pair of vertically opposite angles. We saw, of course, that vertically opposite angles are equal. So, in our diagram, these two angles are equal to one another in size. And we saw that this can help us to solve problems involving vertically opposite angles.