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This page sets out the method for angles in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Angles topic page.
When you see a missing angle, decide which shape or line it belongs to, then subtract the known angles from the total.
When you see two equal sides marked (dashes) or the word isosceles, use the equal base angles.
A full turn is 360°. A straight line is 180°.
A right angle is 90°.
Angles that meet at a point add to 360°.
Subtract the angles you know from 360°.
Two angles that make a straight line add to 180°.
If one is 40°, the other is 140°.
The three interior angles add to 180°.
Find the missing one by subtracting the two you know.
When two lines cross, opposite angles are equal.
The neighbouring angle is then on a straight line with it.
A right-angle mark means 90°, even if the number is not written.
Do not assume an angle is 90° without the mark or a reason.
Angles around a point fill one complete turn, with no gap and no overlap, so they add to 360^°. If every angle but one is known, add the known angles and subtract from 360. Suppose four angles meet at a point and three of them are 80^°, 95^° and 70^°. Their sum is 80+95+70=245, so the fourth angle is 360-245=115^°. Writing the addition first makes the subtraction hard to get wrong.
Use 360^° only when the angles really do go all the way round the same point. Two angles that sit together on a straight line make 180^°, even if other angles are drawn nearby. A common mix-up is to subtract from 180 for a full turn, or from 360 for a line. Look at the vertex, trace a finger around it, and say which total you are using before you calculate.
Method: 1. Add the known angles, then subtract the total from 360^°. 2. For 80^°, 95^° and 70^°, work out 80+95+70=245. 3. The missing angle is 360-245=115^°.
A straight line is half a full turn, so adjacent angles on that line add to 180^°. They share a vertex and sit next to each other. If one angle is 47^°, its neighbour is 180-47=133^°. You can do this without a calculator by going via a friendly number: 180-50=130, then add back the extra 3 you took away, which gives 133. The two angles together rebuild the straight line.
More than two angles can share the same line. Three angles on a straight line still add to 180^°. If two of them are 40^° and 65^°, then 40+65=105 and the third is 180-105=75^°. Do not include an angle that is off the line. A vertically opposite angle equals the given angle; it is not found by subtracting from 180. Name the angle you want, then choose the fact that matches its position.
Method: 1. Add the angles that share the straight line, then subtract from 180^°. 2. Next to 47^°, work out 180-47=133^°. 3. With 40^° and 65^° already used, the third angle is 180-105=75^°.
The interior angles of any triangle add to 180^°. That is true for scalene, isosceles and right-angled triangles. The word interior matters: an angle drawn outside the triangle is not part of this sum. Given two interior angles, add them and subtract from 180. If they are 58^° and 73^°, then 58+73=131 and the third angle is 180-131=49^°. A quick check is that all three numbers add back to 180.
A right angle inside a triangle is 90^° only when the square mark is there. The other two angles then add to 180-90=90^°. If one of them is 35^°, the last is 90-35=55^°. An isosceles triangle has two equal angles, often shown with matching ticks on the sides. If the unequal angle is 40^°, the two equal angles share 180-40=140, so each is 140÷ 2=70^°. The ticks are the evidence; a sketch that merely looks symmetric is not.
Method: 1. Add the two known interior angles and subtract from 180^°. 2. For 58^° and 73^°, work out 58+73=131, then 180-131=49^°. 3. In a right-angled triangle, the other two angles add to 90^°.
When two straight lines cross, they make an X shape with two pairs of vertically opposite angles. Each pair is equal. Here is why. If one angle is 68^°, the adjacent angle on the straight line is 180-68=112^°. The next angle around the point is adjacent to that 112^°, so it is 180-112=68^°. That angle sits opposite the first one, and it matches. Opposite means equal. Adjacent on a line means subtract from 180.
The two obtuse angles match as well, so both are 112^°. Around the point the four angles are 68^°, 112^°, 68^° and 112^°, and 68+112+68+112=360. If that 112^° angle is also an angle of a triangle which already contains 35^°, the third angle of the triangle is 180-112-35=33^°. Read the diagram in order: crossing first, then the triangle. Do not assume any of those angles is a right angle unless a square is drawn.
Method: 1. Copy the angle straight across the crossing: opposite angles are equal. 2. From 68^°, the angle on the straight line is 180-68=112^°. 3. The next angle round the point returns to 180-112=68^°.
Two more facts turn up all the time. In an isosceles triangle two sides are equal, and the two angles opposite those sides, called the base angles, are equal too. If the angle at the top is 44^°, the other two share 180^°-44^°=136^°, so each is 68^°. The angles in any quadrilateral add up to 360^°.
Exam diagrams mark equal sides with small dashes. Look for them before you start. If two sides have the same dash, the angles opposite them are equal. A quadrilateral question usually gives three angles and asks for the fourth: add the three, then take the total from 360^°. Write each step down so the method mark is secure.
Method: 1. Look for equal sides marked with dashes, and for right angle squares. 2. In an isosceles triangle, write down the equal base angles. 3. For a quadrilateral, add the known angles and subtract from 360^°. 4. Check that your angles add to the right total.
Tip: A triangle adds to 180 degrees and a quadrilateral adds to 360 degrees. Say the number aloud as you start.
Many questions give angles as expressions, such as x, 2x and 3x on a straight line. Add the expressions, put the total equal to 180^°, and solve. Here x+2x+3x=6x=180, so x=30. The angles are 30^°, 60^° and 90^°. Always find the angles themselves if the question asks for them. It is a good habit to check by adding the finished angles to see that they make the right total.
Other questions say one angle is a number of times another, such as angle ADC=5× angle CDB on a straight line. Call the small angle a, so the large angle is 5a. The total is 6a=180, so a=30 and the large angle is 150^°. Read what is being asked, because the answer is often the larger angle rather than the letter.
Method: 1. Write each angle as an expression using the letter. 2. Add them and set the sum equal to 180^°, 360^° or whatever the rule gives. 3. Solve the equation to find the letter. 4. Substitute back to find the angle you were asked for and check the total.
Many questions say give a reason for your answer. A reason is a short geometric fact, not a description. Good reasons are: angles on a straight line add up to 180^°; angles around a point add up to 360^°; angles in a triangle add up to 180^°; vertically opposite angles are equal; base angles of an isosceles triangle are equal.
Write the reason straight after the calculation it supports. If the question says give a reason for each stage, every line of working needs its own reason. Do not write because it looks equal, or because it is a Z shape. Use the exact wording of the fact. If a step uses a diagram fact, such as the equal sides marked with dashes, say that too.
Method: 1. Write the angle you are finding, using the letters on the diagram. 2. Do the calculation on the next line. 3. Write a reason that names the angle fact you used. 4. Repeat for each stage, using the answer from the last stage if needed.
Some diagrams need two or three steps. A common one has a triangle with a side extended to make a straight line. First find the missing angle inside the triangle, then use the straight line to find the angle outside. Write each angle on the diagram as you find it, so later steps can use it.
The words Not drawn accurately mean the diagram cannot be measured. A corner that looks like a right angle is not one unless it is marked. Never measure with a protractor and never guess from the picture. Use only the numbers given and the angle facts. If a diagram looks wrong, trust the facts and the numbers.
Method: 1. Mark every given angle and every equal side on the diagram. 2. Find the easiest missing angle first, usually inside a triangle or on a straight line. 3. Use that new angle to find the next one, writing a reason each time. 4. Check the final answer against the picture to see if it is sensible, but trust the working.
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