### Video Transcript

In the diagram below, π΄πΈπΆ is a
right-angled triangle, π΄π΅ is equal to π΄πΆ, and πΆ lies on π΅π·. Calculate the size of π΄πΆπΈ. You must give a reason for each
step of your working.

Our aim in this question is to
calculate the size of angle π΄πΆπΈ, labelled π₯ on the diagram. Our first step is to consider the
isosceles triangle π΄π΅πΆ. This is isosceles as length π΄π΅ is
equal to π΄πΆ. And any triangle with two equal
sides is isosceles.

Any isosceles triangle also has two
equal angles. In this case, angle π΄π΅πΆ is equal
to angle π΅πΆπ΄. Theyβre labelled π¦ on the
diagram. We know that the angles in any
triangle add up to 180 degrees.

In this case, in triangle π΄π΅πΆ,
36 plus π¦ plus π¦ equals 180. Grouping the π¦s by simplifying
gives us 36 plus two π¦ equals 180. Subtracting 36 from both sides of
this equation using the balancing method gives us two π¦ equals 144, as 180 minus 36
is equal to 144. Finally, dividing both sides of
this equation by two gives us π¦ is equal to 72. 144 divided by two, or a half of
144, is equal to 72. This means that angles π΄π΅πΆ and
πΆπ΅π΄ are equal to 72 degrees, as shown on the diagram.

Our final step uses the angle
property that angles on a straight line add up to 180 degrees. In this case, 72 plus π₯ plus 41
equals 180, where π₯ is angle π΄πΆπΈ. 72 plus 41 is equal to 113. Therefore, weβre left with π₯ plus
113 equals 180. Subtracting 113 from both sides of
this equation gives us π₯ is equal to 67, as 180 minus 113 equals 67. We can therefore say that angle
π΄πΆπΈ is equal to 67 degrees.