### Video Transcript

In the diagram, π΄π΅π·πΈπΉ is a
pentagon and π΅πΆπ· is a triangle. π΄π΅πΆ and πΆπ·πΈ are straight
lines. The lengths π΅πΆ and πΆπ· 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¦.

Weβre told in the question that the
lengths π΅πΆ and πΆπ· are equal. This means that triangle π΅πΆπ· 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΅πΆ is equal
to angle πΆπ·π΅.

We know from the diagram that angle
πΆπ·π΅ was equal to 70 degrees. Therefore, angle π·π΅πΆ 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΅πΆπ· 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·πΈ. Weβre told in the question that
π΄π΅πΆ and πΆπ·πΈ 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·πΈ. 70 plus angle π΅π·πΈ equals
180. Once again, subtracting 70 from
both sides of this equation gives us a value for angle π΅π·πΈ of 110 degrees.

We now need to consider the
pentagon π΄π΅π·πΈπΉ 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, youβve 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.