Three straight lines are shown. Work out the value of 𝑥.
Let’s have a look at the diagram that we’ve been given more closely. We see that we’ve been given expressions for the sizes of two of the angles in the diagram and the letter 𝑥 has been used in these expressions. We also notice that two of the lines have these arrow markers on them which indicate that these two lines are parallel to one another.
We’re, therefore, going to need to use facts about angles in parallel lines to answer this question. Now, I’m going to show you two methods that we could use in order to work out the value of 𝑥. The first method is to notice that the two angles which have been marked are an example of co-exterior angles. Exterior means they are on the outside of the two parallel lines. But they’re on the same side of the line that cuts through these parallel lines.
A key fact about co-exterior angles in parallel lines is that they sum to 180 degrees. So we can add together the expressions we’ve been given for these two angles and form an equation. We have three 𝑥 plus 10 degrees plus 86 degrees plus 𝑥 is equal to 180 degrees.
We can now simplify our equation by grouping like terms. We have three 𝑥 plus 𝑥 which gives four 𝑥 then positive 10 plus 86 which gives 96 degrees. So our equation simplifies to four 𝑥 plus 96 degrees is equal to 180 degrees.
To solve this equation for 𝑥, we first need to subtract 96 degrees from each side. On the left-hand side, four 𝑥 plus 96 degrees minus 96 degrees just gives four 𝑥. And on the right-hand side, we know that if we subtract 90 from 180, we get 90. So if we subtract a further six, we get 84. Our equation, therefore, becomes four 𝑥 equals 84 degrees.
To find the value of 𝑥, we need to divide both sides of the equation by four. Now, you may be able to divide 84 by four in your head. But if not, you can do a quick short division method. Eight divided by four is two and four divided by four is one. So 84 divided by four is 21. This tells us that the value of 𝑥 is 21 degrees.
Now, I said we’re going to look at two methods. And the second method will be useful if you didn’t recognise the two angles as co-exterior angles. Instead, you might notice that the angle of three 𝑥 plus 10 degrees is on a straight line with another angle, the angle that I’ve labelled with a pink question mark.
We know that angles on any straight line will always sum to 180 degrees. So we can work out the size of this angle marked with a question mark by subtracting three 𝑥 plus 10 degrees from 180 degrees. That gives 170 degrees minus three 𝑥.
Now, let’s consider the relationship between this angle we’ve just found and the other angle of 86 degrees plus 𝑥. We see that these two angles are enclosed within an F shape. It’s sort of upside down and back to front, but it’s still an F shape.
The proper name for F angles is corresponding angles. And a key fact about corresponding angles is that they are equal. We can, therefore, form an equation by setting the expressions for these two angles equal to one another. We have 86 degrees plus 𝑥 equals 170 degrees minus three 𝑥. We now need to solve this equation to find the value of 𝑥.
First, we note that there are 𝑥 terms on both sides of the equation and we want to collect them all on the same side. As the number of 𝑥s on the right-hand side is negative, we will collect the 𝑥 terms on the left-hand side which we do by adding three 𝑥 to both sides of this equation. This gives 86 degrees plus four 𝑥 is equal to 170 degrees.
The next step is to subtract 86 degrees from each side of the equation. On the left-hand side, we’re left with four 𝑥. And on the right-hand side, if we subtract 70 from 170, we have 100 and then we need to subtract a further 16, which takes us to 84. So we have four 𝑥 equals 84 degrees.
Notice that we have now arrived at the same equation as we had during our first method. So we would solve in exactly the same way by dividing both sides of the equation by four to give 𝑥 equals 21 degrees. You can use either of these two methods, depending on which type of angles in parallel lines you’re most familiar with.
We found that the value of 𝑥 is 21 degrees.