In the diagram, 𝐴𝐵𝐷𝐸𝐹 is a
pentagon and 𝐵𝐶𝐷 is a triangle. 𝐴𝐵𝐶 and 𝐶𝐷𝐸 are straight
lines. The lengths 𝐵𝐶 and 𝐶𝐷 are the
same. Part a) Find the size of angle
𝑥. Give a reason for your answer. Part b) Calculate the size of angle
We’re told in the question that the
lengths 𝐵𝐶 and 𝐶𝐷 are equal. This means that triangle 𝐵𝐶𝐷 is
an isosceles triangle, as any triangle with two equal length sides is isosceles. An isosceles triangle also has two
equal angles. In this case, angle 𝐷𝐵𝐶 is equal
to angle 𝐶𝐷𝐵.
We know from the diagram that angle
𝐶𝐷𝐵 was equal to 70 degrees. Therefore, angle 𝐷𝐵𝐶 must also
be equal to 70 degrees. As this angle was labelled 𝑥, we
can say that 𝑥 is equal to 70 degrees because triangle 𝐵𝐶𝐷 was isosceles.
The second part of our question
asked us to calculate the size of angle 𝑦. In order to do this, we firstly
need to calculate angles 𝐴𝐵𝐷 and 𝐵𝐷𝐸. We’re told in the question that
𝐴𝐵𝐶 and 𝐶𝐷𝐸 are straight lines. We also know that angles on a
straight line sum or add up to 180 degrees.
This means that 70 plus angle
𝐴𝐵𝐷 must equal 180. Subtracting 70 from both sides of
this equation gives us a value for angle 𝐴𝐵𝐷 of 110 degrees. We can use the same method to
calculate the value of angle 𝐵𝐷𝐸. 70 plus angle 𝐵𝐷𝐸 equals
180. Once again, subtracting 70 from
both sides of this equation gives us a value for angle 𝐵𝐷𝐸 of 110 degrees.
We now need to consider the
pentagon 𝐴𝐵𝐷𝐸𝐹 to help us calculate the size of angle 𝑦. Angles in a pentagon sum or add up
to 540 degrees. We can prove this by splitting any
pentagon into the minimum number of triangles.
The least number of triangles a
pentagon can be split into is three. And we know that angles in a
triangle add up to 180 degrees. 180 multiplied by three is equal to
540. Therefore, the angles in a pentagon
add up to 540 degrees.
We can use this same method to
calculate the sum of the interior angles of any polygon. We can split a hexagon into four
triangles. Therefore, the angles in a hexagon
would add up to 180 multiplied by four. This is equal to 720 degrees.
If in an exam, you’ve forgotten
what the angles in any polygon add up to, it is worth doing a little diagram as we
have done here. As the five angles in the pentagon
add up to 540, we can write the following equation: 𝑦 plus 𝑦 plus 110 plus 110
plus 90 is equal to 540.
Simplifying the equation by
grouping like terms gives us two 𝑦 plus 310 equals 540 as 𝑦 plus 𝑦 is equal to
two 𝑦 and 110 plus 110 plus 90 is equal to 310. Subtracting 310 from both sides of
this equation gives us two 𝑦 is equal to 230 as 540 minus 310 is equal to 230. Finally, dividing both sides of
this equation by two gives us a value for 𝑦 of 115 degrees. 230 divided by two is equal to
We can check this answer by adding
the five angles: 115 plus 115 plus 110 plus 110 plus 90 is equal to 540.