Using Triangle and Quadrilateral Properties

You already know that a triangle is worth one hundred and eighty degrees, and that fact hands you the quadrilateral rule for free. Draw one diagonal across any quadrilateral and it splits into two triangles. The diagonal does not create any new angle and it does not lose any either; it simply shares the four original corners between the two triangles. So the four angles must be worth two lots of one hundred and eighty, which is three hundred and sixty degrees. Check it against a rectangle: four right angles, and four times ninety really is three hundred and sixty. Count the sides before you choose which total to use.

Finding a missing angle in a quadrilateral works exactly like a triangle, just with a different total. Add the three angles you have been given, then take that total away from three hundred and sixty. Here the known angles are one hundred and ten, seventy-five and ninety-five, which add to two hundred and eighty. Three hundred and sixty minus two hundred and eighty leaves eighty, so x is eighty degrees. Then check by adding all four back up: one hundred and ten plus seventy-five plus ninety-five plus eighty is three hundred and sixty. That check takes five seconds and it catches almost every arithmetic slip you could make.

Once you know which shape you are dealing with, its properties do most of the work. A parallelogram has two useful facts. Opposite angles are equal, and any two neighbouring angles add to one hundred and eighty. Put those together and a single angle unlocks all four. If one angle is seventy, the angle facing it across the shape is also seventy, and each of its two neighbours is one hundred and eighty minus seventy, which is one hundred and ten. Add them up as a check and you get three hundred and sixty. A rectangle is quicker still, because every one of its angles is ninety degrees.

Some questions give you clues instead of a picture and ask you to name the shape. Work through the clues one at a time and cross shapes off as you go. If you are told there are two pairs of parallel sides, the trapezium and the kite are already out, because a trapezium has only one pair and a kite has none. If you are then told all four sides are equal, you are left with the rhombus and the square. A final clue about right angles decides between those two: with them it is a square, without them it is a rhombus. Each clue narrows the list.

Geometry answers are marked on your reason as much as on your number. Writing just x equals eighty picks up only part of the credit, even when it is right. Writing one hundred and thirty plus sixty plus ninety equals two hundred and eighty, so x equals three hundred and sixty minus two hundred and eighty, which is eighty degrees, with the words angle sum of a quadrilateral in brackets, picks up all of it. The format is short: one line of working, then the reason in brackets. There are only a handful of reasons in this chapter, so learn the exact wording and use it rather than explaining your thinking in a paragraph.