# Video: GCSE Mathematics Foundation Tier Pack 4 • Paper 2 • Question 13

GCSE Mathematics Foundation Tier Pack 4 • Paper 2 • Question 13

02:50

### Video Transcript

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.