Angle Relationships

This whole chapter rests on four facts. A right angle is 90 degrees, and two angles that together make one are called complementary. A straight line is 180 degrees, and two angles making one are supplementary. A full turn is 360 degrees, so angles around a point add to that. And where two lines cross in an X, the angles opposite each other are equal — those are called vertically opposite. That is the entire list. You are not memorising dozens of rules; everything else in this chapter is these four facts applied to a particular picture.

Complementary angles add to 90 degrees, supplementary angles add to 180. The method is identical in both cases — subtract from the total — so the only real decision is which total to use. Look at the picture: a right-angle marker tells you 90, and a straight line tells you 180. If one complementary angle is 35, the other is 90 minus 35, which is 55. If one supplementary angle is 110, the other is 180 minus 110, which is 70. A useful sense-check is that complementary angles must both be acute, under 90 degrees. If your complement comes out as 130, you used the wrong total.

The same idea extends to more than two angles. Angles around a point fill a full turn, so they add to 360 — if two of them are 120 and 95, then x is 360 minus 215, which is 145. Angles sitting on one side of a straight line add to 180, so with 40 and 65 given, y is 75. The method is the same every time: add what you know, then subtract from the total. Write the equation out rather than doing it in your head, because that is what earns method marks. To choose between 180 and 360, ask whether there is a straight line through the point. Angles filling a whole turn use 360; angles on one side of a line use 180.

Where two straight lines cross, the angles opposite each other are equal — these are vertically opposite angles. If one is 70, the angle across from it is also 70, and the two side angles are each 180 minus 70, which is 110. All four add to 360, as they must. Why are opposite angles equal? Because both sit on the same straight line as the same neighbour: if a plus 70 is 180 and a plus c is 180, then c has to be 70. One warning about the name: vertically opposite has nothing to do with up and down. It means across the vertex, the crossing point, so left-and-right pairs count just as much.

One habit separates full marks from half marks in geometry: giving a reason. Every line of working should name the rule you used, in brackets — 'a equals 65 degrees, angles on a straight line add to 180', or 'b equals 65, vertically opposite angles are equal'. In geometry the reason is worth as many marks as the number itself, so a bare 65 gets one mark while 65 with its reason gets both. It takes about four words. And it has a second benefit: writing the reason forces you to check that the rule genuinely applies to the picture in front of you, which catches errors before they cost you.