Video: AQA GCSE Mathematics Higher Tier Pack 5 β€’ Paper 2 β€’ Question 8

𝐴𝐡𝐢𝐷 is a parallelogram. 𝐴𝐷 = 𝐴𝑆. 𝐷𝑆 = 𝑆𝐡. Work out the size of angle π‘₯.

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Video Transcript

𝐴𝐡𝐢𝐷 is a parallelogram. 𝐴𝐷 equals 𝐴𝑆. 𝐷𝑆 equals 𝑆𝐡. Work out the size of angle π‘₯.

As the shape 𝐴𝐡𝐢𝐷 is a parallelogram, we know that the sides 𝐴𝐷 and 𝐡𝐢 are parallel. Likewise, 𝐴𝐡 and 𝐷𝐢 are parallel. In any question like this involving our angle properties, it is worth working out one angle at a time and then labelling that angle on the diagram.

Triangle 𝐴𝐷𝑆 is isosceles, as the length of 𝐴𝑆 is equal to the length 𝐴𝐷. This means that the angle 𝐴𝐷𝑆 is equal to the angle 𝐴𝑆𝐷. We were told that angle 𝐴𝑆𝐷 is equal to 26 degrees. Therefore, angle 𝐴𝐷𝑆 is also equal to 26 degrees.

The angles in any triangle add up to 180 degrees. This means that angle 𝐷𝐴𝑆 plus angle 𝐴𝐷𝑆 plus angle 𝐴𝑆𝐷 equals 180. We already know that two of the angles in the triangle equal 26 degrees. Therefore, angle 𝐷𝐴𝑆 plus 26 plus 26 is equal to 180. 26 plus 26 is equal to 52. Subtracting 52 from both sides of this equation gives us angle 𝐷𝐴𝑆 is equal to 128 degrees, as 180 minus 52 is equal to 128.

We have now worked out all three angles in the triangle 𝐷𝐴𝑆. 𝐴𝑆𝐷 is 26 degrees, 𝐴𝐷𝑆 is 26 degrees, and 𝐷𝐴𝑆 is 128 degrees.

Angles on a straight line also have a sum of 180 degrees. This means that angle 𝐴𝑆𝐷 and angle 𝐷𝑆𝐡 equal 180. We know that angle 𝐴𝑆𝐷 is 26 degrees. Subtracting 26 from both sides of this equation gives us angle 𝐷𝑆𝐡 is equal to 154 degrees, as 180 minus 26 is equal to 154.

The triangle 𝐡𝐷𝑆 is also isosceles as 𝐷𝑆 is equal to 𝑆𝐡. This means that the angles 𝑆𝐷𝐡 and 𝑆𝐡𝐷 are equal. As the angles in a triangle once again add up to 180 degrees, we can calculate these angles by subtracting 154 from 180 and then dividing by two. 180 minus 154 is equal to 26. And 26 divided by two is equal to 13. Therefore, angle 𝑆𝐷𝐡 and angle 𝑆𝐡𝐷 are equal to 13 degrees.

We have now calculated all three angles in the triangle 𝐡𝐷𝑆. 𝑆𝐷𝐡 is equal to 13 degrees, 𝑆𝐡𝐷 is equal to 13 degrees, and 𝐷𝑆𝐡 is equal to 154 degrees. At this point, we have a few options to help us calculate the size of angle π‘₯. We could use the fact that opposite angles in a parallelogram are equal. This means that angle 𝐷𝐴𝑆 is equal to angle 𝐡𝐢𝐷. They are both equal to 128 degrees.

An easier way would be to use the fact that cointerior or C-angles add up to 180 degrees. As the lines 𝐴𝐡 and 𝐷𝐢 are parallel, angle 𝐷𝐴𝑆 and angle 𝐴𝐷𝐢 equal 180 degrees. We know that angle 𝐷𝐴𝑆 is 128 degrees. This means we can subtract 128 from both sides of the equation, giving us angle 𝐴𝐷𝐢 is equal to 52 degrees. Angle 𝐴𝐷𝐢 is made up of three smaller angles: 26 degrees, 13 degrees, and π‘₯ degrees. This gives us an equation that we can solve for π‘₯. 26 plus 13 plus π‘₯ is equal to 52. 26 plus 13 is equal to 39. We can then subtract 39 from both sides of this equation. 52 minus 39 is equal to 13. This means that the size of angle π‘₯ is 13 degrees.

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