𝐴𝐵, 𝐶𝐷, and 𝐸𝐹 are straight
lines. 𝐴𝐵 is parallel to 𝐶𝐷. Find the value of 𝑦.
We can see that 𝑦 has been used in
an expression for the size of one of the angles in this diagram. We were told in the question that
the line 𝐴𝐵 is parallel to the line 𝐶𝐷. So in order to answer this
question, we’re going to need to use some facts about angles in parallel lines. Notice that the expressions we’ve
been given for two other angles in this diagram are in terms of a second variable
𝑥. So we’re probably going to need to
calculate the value of 𝑥 before we can calculate the value of 𝑦. I’m going to introduce the letters
𝐺 and 𝐻 onto the diagram to represent the points where the line 𝐸𝐹 crosses 𝐴𝐵
Let’s look first of all at the
angles 𝐴𝐺𝐻 and 𝐶𝐻𝐸. And we notice that they’re enclosed
within an 𝐹 shape. It’s backwards. But it’s still an 𝐹 shape. The proper term for angles which
are enclosed within an 𝐹 shape is corresponding angles. So we can conclude that angle
𝐴𝐺𝐻 and 𝐶𝐻𝐸 are corresponding angles. Corresponding angles are equal to
each other. So angle 𝐴𝐺𝐻 is equal to angle
𝐶𝐻𝐸. This means that we can add the
expression of two 𝑥 plus 27 degrees to the top part of our diagram.
Now let’s look at angles 𝐴𝐺𝐹 and
𝐴𝐺𝐻. That’s the angle I have marked in
pink and the angle I have marked in orange. These two angles lie on a straight
line. And we know that the sum of angles
on a straight line is 180 degrees. So angle 𝐴𝐺𝐹 plus angle 𝐴𝐺𝐻
equals 180 degrees. We can form an equation by adding
the expressions for these two angles and setting it equal to 180. We have four 𝑥 plus three plus two
𝑥 plus 27 equals 180.
We can simplify this equation by
grouping like terms. Firstly, four 𝑥 plus two 𝑥 is six
𝑥 and three plus 27 is 30. We are now in a position to be able
to solve this equation. The first step is to subtract 30
from each side of the equation. On the left-hand side, we’re left
with six 𝑥 and on the right-hand side, 180 minus 30 is 150.
Next, we need to divide both sides
of this equation by six. On the left-hand side six 𝑥
divided by six is just 𝑥 and on the right-hand side, 150 divided by six is 25. You can work this out with a bit of
logic if you remember that four lots of 25 are 100. So six lots of 25 are 150 or you
can use a short division method. There are no sixes is in one. So we carry the one into the next
column. There are two sixes in 15 as two
sixes are 12 with a remainder of three. And there are five sixes in 30 with
no remainder, giving our answer of 25.
So we’ve found the value of 𝑥. But how would this help us with the
question which was to find the value of 𝑦? There are two ways that we could
now work out the value of 𝑦. Firstly, we could note that angle
𝐴𝐺𝐻 and 𝐵𝐺𝐻 are on a straight line, which means the sum of these two angles
must be 180 degrees. As we know the value of 𝑥, we can
work out the size of angle 𝐴𝐺𝐻 and then form an equation to find the value of
𝑦. Or we can use the fact that angles
𝐴𝐺𝐹 and 𝐵𝐺𝐻 are vertically opposite angles. They’re formed by the intersection
of two straight lines. And we know that vertically
opposite angles are equal. Let’s use this method.
Angle 𝐴𝐺𝐹 is four 𝑥 plus three
degrees. And we know the value of 𝑥 is
25. So we can work out the size of this
angle by substituting 25 for 𝑥. Four multiplied by 25 is 100 and
adding three gives 103. So angle 𝐴𝐺𝐹 is 103 degrees. But as vertically opposite angles
are equal, angle 𝐵𝐺𝐻 is also equal to 103 degrees. So we can form an equation: five 𝑦
minus two equals 103.
To solve for 𝑦, we must first add
two to each side, giving five 𝑦 equals 105, and then divide both sides of the
equation by five to give 𝑦 equals 21. You can see that 105 divided by
five is 21, either using a short division method or remembering that five times 20
is 100. So five times 21 will be 105.
We found the value of 𝑦. 𝑦 is equal to 21.