Find unknown angles

Now the same three facts, but the answer no longer falls out of a single subtraction. Four shapes of question come up again and again. A **right angle** is one of the angles given, so 90° comes off the line before anything else does — which is the one place in this topic where subtracting from 90 is right rather than wrong. An unknown is written as a **multiple**, like x and 2x, so you count the angles in units, take the given angles off, and divide. A remainder is **shared between equal angles**, so you find the leftover first and only then divide it. And sometimes the whole figure is given as a **ratio**, 4x : 7x : 9x, with no number in it anywhere. In every one of them the order matters: subtract first, divide second. Dividing the whole 180° or 360° before taking off what you were given is the mistake this lesson exists to break.

Worked example

Problem The angles at a point are 150°, x and 2x. Find x, and then find the angle marked 2x.

The rays go all the way round the point, so everything adds to 360°.
Take off the angle you are given: 360 − 150 = 210. That 210° is shared between x and 2x.
Count them in units: 1 unit + 2 units = 3 units. So 3 units = 210.
One unit is 210 ÷ 3 = 70°, so x = 70°, and 2x = 2 × 70 = 140°.
Watch out 360 ÷ 3 = 120 is the answer to a different question. The 150° has to come off BEFORE you divide — subtract first, divide second.
Tip Always check by putting your answer back into the figure: 150 + 70 + 140 = 360. If the angles do not add back up to 180° or 360°, something has gone wrong.
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