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

π΄π΅πΆπ· is a parallelogram. π΄π· = π΄π. π·π = ππ΅. Work out the size of angle π₯.

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### Video Transcript

π΄π΅πΆπ· is a parallelogram. π΄π· equals π΄π. π·π equals ππ΅. Work out the size of angle π₯.

As the shape π΄π΅πΆπ· is a parallelogram, we know that the sides π΄π· and π΅πΆ are parallel. Likewise, π΄π΅ and π·πΆ 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 π΄π·π is isosceles, as the length of π΄π is equal to the length π΄π·. This means that the angle π΄π·π is equal to the angle π΄ππ·. We were told that angle π΄ππ· is equal to 26 degrees. Therefore, angle π΄π·π is also equal to 26 degrees.

The angles in any triangle add up to 180 degrees. This means that angle π·π΄π plus angle π΄π·π plus angle π΄ππ· equals 180. We already know that two of the angles in the triangle equal 26 degrees. Therefore, angle π·π΄π 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 π·π΄π is equal to 128 degrees, as 180 minus 52 is equal to 128.

We have now worked out all three angles in the triangle π·π΄π. π΄ππ· is 26 degrees, π΄π·π is 26 degrees, and π·π΄π is 128 degrees.

Angles on a straight line also have a sum of 180 degrees. This means that angle π΄ππ· and angle π·ππ΅ equal 180. We know that angle π΄ππ· is 26 degrees. Subtracting 26 from both sides of this equation gives us angle π·ππ΅ is equal to 154 degrees, as 180 minus 26 is equal to 154.

The triangle π΅π·π is also isosceles as π·π is equal to ππ΅. This means that the angles ππ·π΅ and ππ΅π· 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 ππ·π΅ and angle ππ΅π· are equal to 13 degrees.

We have now calculated all three angles in the triangle π΅π·π. ππ·π΅ is equal to 13 degrees, ππ΅π· is equal to 13 degrees, and π·ππ΅ 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 π·π΄π is equal to angle π΅πΆπ·. 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 π΄π΅ and π·πΆ are parallel, angle π·π΄π and angle π΄π·πΆ equal 180 degrees. We know that angle π·π΄π is 128 degrees. This means we can subtract 128 from both sides of the equation, giving us angle π΄π·πΆ is equal to 52 degrees. Angle π΄π·πΆ 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.