GCSE Maths: Angles on a line, around a point and vertically opposite (Foundation)
Part of the GCSE Maths angles, polygons and symmetry chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Angles on a line, around a point and vertically opposite
Three angle facts are the foundation of everything in this chapter. You will use them in almost every question, so learn them well.
Angles on a straight line add up to 180°. A straight line is a half turn, and a half turn is 180°. If two or more angles sit side by side on a straight line, their sizes make 180° altogether. For example, if one angle on a line is 112°, the angle next to it must be 180° − 112° = 68°.
Angles around a point add up to 360°. A full turn is 360°. If angles meet at one point and fill all the space around it, they make a full turn, so they add up to 360°.
Vertically opposite angles are equal. When two straight lines cross, they make an X. The angles that face each other across the crossing point are called vertically opposite angles. (“Vertically” here has nothing to do with up and down; it refers to the vertex, the point where the lines cross.) Vertically opposite angles are always equal. Why? Look at the picture: angle a and angle b are both on a straight line with the same angle c, so a = 180° − c and b = 180° − c. They must be equal.
Giving reasons. In an exam you are often told to “give a reason”. A good reason names the angle fact in words, exactly as above. Write the reason next to each step: “x = 68° (angles on a straight line add up to 180°)”. Never write “because it is the same” or “because it looks like it”.
Key ideas
- Angles on a straight line add up to 180°.
- Angles around a point add up to 360°.
- Vertically opposite angles are equal.
- Method: (1) decide which fact fits, (2) add the angles you know, (3) subtract from the total, (4) write the reason.
- Angles on a line are next to each other; vertically opposite angles are opposite each other at an X.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.112 = 112°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 112 = 68°
x = 68°
Check. Add all the angles: 112 + 68 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles around a point must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A full turn is 360°. Angles that meet at a point and fill the space around it share one full turn.Angles around a point add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.95 + 120 + 70 = 285°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 285 = 75°
x = 75°
Check. Add all the angles: 95 + 120 + 70 + 75 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Use angles on a straight line first, then vertically opposite angles.
- Find a: it is next to the 64° angle on a straight line. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
a = 180 − 64 = 116° - Find b: it is opposite the 64° angle. The two angles face each other across the crossing point, so they are vertically opposite.Vertically opposite angles are equal.
b = 64° - Find c: it is opposite a. Again an X shape: c and a are vertically opposite.Vertically opposite angles are equal.
c = a = 116°
Check. Add all four angles around the crossing point: 64 + 116 + 64 + 116 = 360°. ✓
Watch out. The two 64° angles are equal, and the two 116° angles are equal. Opposite angles match; neighbouring angles add to 180°.
Watch out
Check which total to use. A straight line is 180°; a full turn around a point is 360°. If the angles are on a line and you subtract from 360°, you will get a wrong (and usually too large) answer. Also remember to write the reason in words.Exercise 17B Angles on a line, around a point and vertically opposite
Non-calculatorKey words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.112 = 112°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 112 = 68°
x = 68°
Check. Add all the angles: 112 + 68 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.48 = 48°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 48 = 132°
x = 132°
Check. Add all the angles: 48 + 132 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.35 + 90 = 125°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 125 = 55°
x = 55°
Check. Add all the angles: 35 + 90 + 55 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.70 + 40 + 55 = 165°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 165 = 15°
y = 15°
Check. Add all the angles: 70 + 40 + 55 + 15 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles around a point must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A full turn is 360°. Angles that meet at a point and fill the space around it share one full turn.Angles around a point add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.130 + 120 = 250°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 250 = 110°
a = 110°
Check. Add all the angles: 130 + 120 + 110 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles around a point must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A full turn is 360°. Angles that meet at a point and fill the space around it share one full turn.Angles around a point add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.85 + 95 + 110 = 290°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 290 = 70°
b = 70°
Check. Add all the angles: 85 + 95 + 110 + 70 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles around a point must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A full turn is 360°. Angles that meet at a point and fill the space around it share one full turn.Angles around a point add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.72 + 100 + 64 + 58 = 294°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 294 = 66°
c = 66°
Check. Add all the angles: 72 + 100 + 64 + 58 + 66 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.64 = 64°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 64 = 116°
a = 116°
Check. Add all the angles: 64 + 116 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Vertically opposite angles are equal: they are opposite each other where two straight lines cross.
- Say which angle is equal to b. The two lines cross at a point. The angles that face each other across the crossing point are vertically opposite (opposite each other at the X).The 64° angle and b are vertically opposite.
- State the angle fact. This is the reason we give in an exam.Vertically opposite angles are equal.
- Write the answer. The two angles are equal, so they have the same size.b = 64°
Check. Check: b and the 64° angle are opposite each other at the X, so they must look the same size. ✓
Watch out. Do not confuse vertically opposite angles (opposite at the X) with angles on a line (next to each other, adding to 180°).
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Angle c is opposite angle a, which we found in part (a).
- Say which angle is equal to c. The two lines cross at a point. The angles that face each other across the crossing point are vertically opposite (opposite each other at the X).Angle a (116°) and c are vertically opposite.
- State the angle fact. This is the reason we give in an exam.Vertically opposite angles are equal.
- Write the answer. The two angles are equal, so they have the same size.c = 116°
Check. Check with a different fact: c and the 64° angle are on a straight line, so c = 180 − 64 = 116°. ✓
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles on a straight line must add up to 180°. Add the angles we know and subtract from 180°.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add up the angles we already know. We need their total, so we can see how much of the 180° is left.127 = 127°
- Subtract that total from 180°. All the angles together must make 180°, so the missing angle is whatever is left over.180 − 127 = 53°
a = 53°
Check. Add all the angles: 127 + 53 = 180°, which is 180°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Vertically opposite angles are equal: they are opposite each other where two straight lines cross.
- Say which angle is equal to b. The two lines cross at a point. The angles that face each other across the crossing point are vertically opposite (opposite each other at the X).The 127° angle and b are vertically opposite.
- State the angle fact. This is the reason we give in an exam.Vertically opposite angles are equal.
- Write the answer. The two angles are equal, so they have the same size.b = 127°
Check. Check: b and the 127° angle are opposite each other at the X, so they must look the same size. ✓
Watch out. Do not confuse vertically opposite angles (opposite at the X) with angles on a line (next to each other, adding to 180°).
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Angle c is opposite angle a, which we found in part (a).
- Say which angle is equal to c. The two lines cross at a point. The angles that face each other across the crossing point are vertically opposite (opposite each other at the X).Angle a (53°) and c are vertically opposite.
- State the angle fact. This is the reason we give in an exam.Vertically opposite angles are equal.
- Write the answer. The two angles are equal, so they have the same size.c = 53°
Check. Check with a different fact: c and the 127° angle are on a straight line, so c = 180 − 127 = 53°. ✓
Method. Recall the fact.
- Think of a straight line as a half turn. A full turn is 360°, and half of 360 is 180.360 ÷ 2 = 180
- Write the answer. 180
Check. A straight line is a half turn: 180°. ✓
Method. Recall the fact.
- Think of a full turn. Going all the way round a point is one full turn.one full turn = 360°
- Write the answer. 360
Check. A full turn is 360°. ✓
Method. Recall the fact.
- Picture two lines crossing in an X. The angles that face each other across the crossing point are vertically opposite.top and bottom are a pair; left and right are a pair
- Write the answer. Each pair has the same size.equal
Check. In an X, opposite angles match. ✓
Method. Angles on a straight line must add up to 180°. Add them and compare.
- Recall the fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Add the two angles. If they are on a straight line the total must be exactly 180.107 + 83 = 190°
- Compare with 180°. 190 is not 180, so the angles cannot make a straight line.No: 190° ≠ 180°
Check. If the angles were 107° and 73° they would total 180°. ✓
Watch out. A reason must mention the total: “they do not add up to 180°”.
Method. The 12 hours share a full turn of 360°.
- Find the angle for one hour. A full turn is 360° and there are 12 equal gaps.360 ÷ 12 = 30°
- Count the gaps between the hands. At 3 o’clock the hands point at 12 and at 3: 3 gaps.3 × 30 = 90°
Check. 3 o’clock is a quarter turn: 360 ÷ 4 = 90°. ✓
Method. The 12 hours share a full turn of 360°.
- Find the angle for one hour. A full turn is 360° and there are 12 equal gaps.360 ÷ 12 = 30°
- Count the gaps between the hands. At 5 o’clock the hands point at 12 and at 5: 5 gaps.5 × 30 = 150°
Check. 6 gaps would be 180°, a straight line, and 150° is a bit less. ✓
Method. Use the smaller of the two angles between the hands.
- Find the angle for one hour. A full turn is 360° and there are 12 equal gaps.360 ÷ 12 = 30°
- Count the gaps the long way. From 12 round to 8 is 8 gaps.8 × 30 = 240° (a reflex angle)
- Find the smaller angle. The two angles around the point add up to 360°.360 − 240 = 120°
Check. The short way round from 8 to 12 is 4 gaps: 4 × 30 = 120°. ✓
Watch out. Give the smaller angle unless the question says otherwise.
Method. Find the angle of one slice, then find the angle for the slices that are left.
- Find the angle of one slice. The whole pizza is a full turn, 360°, shared equally by 8 slices.360 ÷ 8 = 45°
- Find the number of slices left. Anna takes 3 of the 8.8 − 3 = 5 slices
- Find the angle for 5 slices. Each slice is 45°.5 × 45 = 225°
Check. Anna’s slices take 3 × 45 = 135°, and 135 + 225 = 360°. ✓
Watch out. Check that your angles add up to 360° round the centre.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 180°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Write an equation. The angles must add up to 180°, so we add all the expressions and put the answer equal to 180.x° + 2x° = 180°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.3x = 180
- Divide both sides by 3. 3x means 3 × x, so dividing by 3 leaves x on its own.x = 180 ÷ 3 = 60
- Substitute x = 60 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.x° = 1 × 60 = 60°
2x° = 2 × 60 = 120°
Check. Add the angles: 60 + 120 = 180°, which is 180°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 180°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Write an equation. The angles must add up to 180°, so we add all the expressions and put the answer equal to 180.(3x + 20°) + (2x + 10°) = 180°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.5x + 30 = 180
- Get the number on the other side. Subtract 30 from both sides.5x = 150
- Divide both sides by 5. 5x means 5 × x, so dividing by 5 leaves x on its own.x = 150 ÷ 5 = 30
- Substitute x = 30 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.3x + 20° = 3 × 30 + 20 = 110°
2x + 10° = 2 × 30 + 10 = 70°
Check. Add the angles: 110 + 70 = 180°, which is 180°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 360°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. A full turn is 360°. Angles that meet at a point and fill the space around it share one full turn.Angles around a point add up to 360°.
- Write an equation. The angles must add up to 360°, so we add all the expressions and put the answer equal to 360.2x° + (x + 30°) + (3x − 30°) = 360°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.6x = 360
- Divide both sides by 6. 6x means 6 × x, so dividing by 6 leaves x on its own.x = 360 ÷ 6 = 60
- Substitute x = 60 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.2x° = 2 × 60 = 120°
x + 30° = 1 × 60 + 30 = 90°
3x − 30° = 3 × 60 − 30 = 150°
Check. Add the angles: 120 + 90 + 150 = 360°, which is 360°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Method. Check which fact applies to angles on a straight line.
- Recall the fact for a straight line. A straight line is half a turn, which is 180°. Angles that sit side by side on a line share that half turn.Angles on a straight line add up to 180°.
- Spot the mistake. Dan used 360°, which is the total for a full turn around a point, not for a straight line.wrong total: 360 instead of 180
- Do the correct calculation. Subtract from 180°.180 − 112 = 68°
Check. 112 + 68 = 180. ✓
Watch out. Ask yourself: is it a straight line (180°) or a full turn (360°)?
Method. Angles AEC and AED are next to each other on the straight line CD, so they add up to 180°.
- Say how the angles are related. Angles on a straight line add up to 180°.angle AEC + angle AED = 180°
- Use one unit for the smaller angle. If AED is 1 part, AEC is 3 parts, so there are 4 parts altogether.1 + 3 = 4 parts = 180°
- Find one part. Divide by 4.180 ÷ 4 = 45°
Check. AEC = 3 × 45 = 135 and 135 + 45 = 180°. ✓
Method. Look at which angle is opposite angle BEC where AB and CD cross.
- Spot the pair. Angle BEC is opposite angle AED because AB and CD are straight lines crossing at E.BEC and AED are vertically opposite
- Use the fact. Vertically opposite angles are equal.angle BEC = angle AED = 45°
Check. BEC and AEC are on a straight line: 45 + 135 = 180°. ✓