# Video: GCSE Mathematics Foundation Tier Pack 1 β’ Paper 3 β’ Question 17

GCSE Mathematics Foundation Tier Pack 1 β’ Paper 3 β’ Question 17

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### Video Transcript

In the diagram, π΄π΅π·πΈπΉ is a pentagon and π΅πΆπ· is a triangle. π΄π΅πΆ and πΆπ·πΈ are straight lines. The lengths π΅πΆ and πΆπ· are the same. Part a) Find the size of angle π₯. Give a reason for your answer. Part b) Calculate the size of angle π¦.

Weβre told in the question that the lengths π΅πΆ and πΆπ· are equal. This means that triangle π΅πΆπ· 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 π·π΅πΆ is equal to angle πΆπ·π΅.

We know from the diagram that angle πΆπ·π΅ was equal to 70 degrees. Therefore, angle π·π΅πΆ must also be equal to 70 degrees. As this angle was labelled π₯, we can say that π₯ is equal to 70 degrees because triangle π΅πΆπ· was isosceles.

The second part of our question asked us to calculate the size of angle π¦. In order to do this, we firstly need to calculate angles π΄π΅π· and π΅π·πΈ. Weβre told in the question that π΄π΅πΆ and πΆπ·πΈ 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 π΄π΅π· must equal 180. Subtracting 70 from both sides of this equation gives us a value for angle π΄π΅π· of 110 degrees. We can use the same method to calculate the value of angle π΅π·πΈ. 70 plus angle π΅π·πΈ equals 180. Once again, subtracting 70 from both sides of this equation gives us a value for angle π΅π·πΈ of 110 degrees.

We now need to consider the pentagon π΄π΅π·πΈπΉ to help us calculate the size of angle π¦. 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, youβ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: π¦ plus π¦ plus 110 plus 110 plus 90 is equal to 540.

Simplifying the equation by grouping like terms gives us two π¦ plus 310 equals 540 as π¦ plus π¦ is equal to two π¦ and 110 plus 110 plus 90 is equal to 310. Subtracting 310 from both sides of this equation gives us two π¦ 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 π¦ of 115 degrees. 230 divided by two is equal to 115.

We can check this answer by adding the five angles: 115 plus 115 plus 110 plus 110 plus 90 is equal to 540.