In the diagram below, 𝐴𝐸𝐶 is a
right-angled triangle, 𝐴𝐵 is equal to 𝐴𝐶, and 𝐶 lies on 𝐵𝐷. Calculate the size of 𝐴𝐶𝐸. You must give a reason for each
step of your working.
Our aim in this question is to
calculate the size of angle 𝐴𝐶𝐸, labelled 𝑥 on the diagram. Our first step is to consider the
isosceles triangle 𝐴𝐵𝐶. This is isosceles as length 𝐴𝐵 is
equal to 𝐴𝐶. And any triangle with two equal
sides is isosceles.
Any isosceles triangle also has two
equal angles. In this case, angle 𝐴𝐵𝐶 is equal
to angle 𝐵𝐶𝐴. They’re labelled 𝑦 on the
diagram. We know that the angles in any
triangle add up to 180 degrees.
In this case, in triangle 𝐴𝐵𝐶,
36 plus 𝑦 plus 𝑦 equals 180. Grouping the 𝑦s by simplifying
gives us 36 plus two 𝑦 equals 180. Subtracting 36 from both sides of
this equation using the balancing method gives us two 𝑦 equals 144, as 180 minus 36
is equal to 144. Finally, dividing both sides of
this equation by two gives us 𝑦 is equal to 72. 144 divided by two, or a half of
144, is equal to 72. This means that angles 𝐴𝐵𝐶 and
𝐶𝐵𝐴 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 𝑥 plus 41
equals 180, where 𝑥 is angle 𝐴𝐶𝐸. 72 plus 41 is equal to 113. Therefore, we’re left with 𝑥 plus
113 equals 180. Subtracting 113 from both sides of
this equation gives us 𝑥 is equal to 67, as 180 minus 113 equals 67. We can therefore say that angle
𝐴𝐶𝐸 is equal to 67 degrees.