Hard GCSE Angles in parallel lines Questions

Challenging, exam-style GCSE Angles in parallel lines questions with worked solutions. Stretch yourself on the hardest vertically opposite angles, co-interior angles, parallel lines, multi-step reasoning problems.

vertically opposite anglesco-interior anglesparallel linesmulti-step reasoningangles at a pointangles on a straight line
GCSE Foundation34 questionsStep-by-step solutions
Question 1
5 markschallenging
ABAB and CDCD are parallel straight lines, with ABAB below CDCD. AA and CC are on the left; BB and DD are on the right. A straight line EFEF crosses ABAB at PP and crosses CDCD at QQ, with EE below ABAB and FF above CDCD. Angle PQD=43∘PQD = 43^\circ. Kian writes angle APQ=43∘APQ = 43^\circ, and then angle BPE=43∘BPE = 43^\circ. Which angle fact does Kian use in his second step?
Show worked solution

Worked solution

  1. Write down what you are given

    ∠APQ=43∘\angle APQ = 43^\circ

    Angle APQ is 43°. The question is which angle fact takes you from there to the angle that was written down.

  2. Look at the first step, which is not the one being asked about

    ∠APQ=43∘\angle APQ = 43^\circ

    The first step uses a different fact (alternate angles are equal) to get angle APQ = 43°. The question asks about the SECOND step only.

  3. Describe where angle APQ and angle BPE sit

    ∠APQ=43∘,∠BPE=43∘\angle APQ = 43^\circ, \quad \angle BPE = 43^\circ

    Angle APQ and angle BPE are opposite each other at the point where two straight lines cross.

  4. Name the fact

    Vertically opposite angles are equal\text{Vertically opposite angles are equal}

    Vertically opposite angles are equal.

  5. Check the fact really does give that answer

    ∠BPE=∠APQ=43∘\angle BPE = \angle APQ = 43^\circ

    Using the fact on angle APQ gives 43°, which is exactly what was written down.

  6. Check the numbers in the whole chain

    43∘→43∘→43∘43^\circ \rightarrow 43^\circ \rightarrow 43^\circ

    Both steps give the numbers that were written down, so the working is right — only the reason for the second step is in question.

  7. Rule out one of the other answers

    Alternate angles are equal\text{Alternate angles are equal}

    "Alternate angles are equal." is a true fact and it happens to give the same number, but it is the wrong reason — angle APQ and angle BPE are not that kind of pair, so it would earn no mark.

  8. Rule out one of the other answers

    Corresponding angles are equal\text{Corresponding angles are equal}

    "Corresponding angles are equal." is a true fact and it happens to give the same number, but it is the wrong reason — angle APQ and angle BPE are not that kind of pair, so it would earn no mark.

  9. Rule out one of the other answers

    Co-interior angles add up to 180∘\text{Co-interior angles add up to 180}^\circ

    "Co-interior angles add up to 180°." is a true fact, but it does not describe this pair: applying it here would give 137°, not 43°.

  10. Rule out one of the other answers

    Vertically opposite angles add up to 180∘\text{Vertically opposite angles add up to 180}^\circ

    "Vertically opposite angles add up to 180°." is simply not true, so it cannot justify anything. Used here it would give 137°.

  11. Compare the two steps

    step 1: Alternate angles are equal,  step 2: Vertically opposite angles are equal\text{step 1: } \text{Alternate angles are equal}, \; \text{step 2: } \text{Vertically opposite angles are equal}

    The two steps use different facts. Naming the first one for the second step is the commonest error here.

  12. Check the fact against the figure once more

    Vertically opposite angles are equal\text{Vertically opposite angles are equal}

    Angle APQ and angle BPE are opposite each other at the point where two straight lines cross.

  13. Write the reason out in full

    Vertically opposite angles are equal\text{Vertically opposite angles are equal}

    A reason like "parallel lines" on its own earns nothing: the fact has to be named. Vertically opposite angles are equal.

  14. Note the mark scheme

    1 mark: correct fact named\text{1 mark: correct fact named}

    One mark is available and it is for naming the correct angle fact.

  15. State the fact that earns the mark

    Vertically opposite angles are equal\text{Vertically opposite angles are equal}

    The reason that must be written down is: Vertically opposite angles are equal.

Answer
Vertically opposite angles are equal\text{Vertically opposite angles are equal}
Question 2
5 markschallenging
ABAB and CDCD are parallel straight lines, with ABAB below CDCD. AA and CC are on the left; BB and DD are on the right. A straight line EFEF crosses ABAB at PP and crosses CDCD at QQ, with EE below ABAB and FF above CDCD. Angle APQ=106∘APQ = 106^\circ. Sara writes angle QPB=74∘QPB = 74^\circ, and then angle PQC=74∘PQC = 74^\circ. Which angle fact does Sara use in her second step?
Show worked solution

Worked solution

  1. Write down what you are given

    ∠QPB=74∘\angle QPB = 74^\circ

    Angle QPB is 74°. The question is which angle fact takes you from there to the angle that was written down.

  2. Look at the first step, which is not the one being asked about

    ∠QPB=74∘\angle QPB = 74^\circ

    The first step uses a different fact (angles on a straight line add up to 180°) to get angle QPB = 74°. The question asks about the SECOND step only.

  3. Describe where angle QPB and angle PQC sit

    ∠QPB=74∘,∠PQC=74∘\angle QPB = 74^\circ, \quad \angle PQC = 74^\circ

    Angle QPB and angle PQC lie between the parallel lines on opposite sides of the transversal — the Z-shape — so they are alternate angles.

  4. Name the fact

    Alternate angles are equal\text{Alternate angles are equal}

    Alternate angles are equal.

  5. Check the fact really does give that answer

    ∠PQC=∠QPB=74∘\angle PQC = \angle QPB = 74^\circ

    Using the fact on angle QPB gives 74°, which is exactly what was written down.

  6. Check the numbers in the whole chain

    74∘→74∘→74∘74^\circ \rightarrow 74^\circ \rightarrow 74^\circ

    Both steps give the numbers that were written down, so the working is right — only the reason for the second step is in question.

  7. Rule out one of the other answers

    Angles on a straight line add up to 180∘\text{Angles on a straight line add up to 180}^\circ

    "Angles on a straight line add up to 180°." is a true fact, but it does not describe this pair: applying it here would give 106°, not 74°.

  8. Rule out one of the other answers

    Co-interior angles add up to 180∘\text{Co-interior angles add up to 180}^\circ

    "Co-interior angles add up to 180°." is a true fact, but it does not describe this pair: applying it here would give 106°, not 74°.

  9. Rule out one of the other answers

    Vertically opposite angles are equal\text{Vertically opposite angles are equal}

    "Vertically opposite angles are equal." is a true fact and it happens to give the same number, but it is the wrong reason — angle QPB and angle PQC are not that kind of pair, so it would earn no mark.

  10. Rule out one of the other answers

    Alternate angles add up to 180∘\text{Alternate angles add up to 180}^\circ

    "Alternate angles add up to 180°." is simply not true, so it cannot justify anything. Used here it would give 106°.

  11. Compare the two steps

    step 1: Angles on a straight line add up to 180∘,  step 2: Alternate angles are equal\text{step 1: } \text{Angles on a straight line add up to 180}^\circ, \; \text{step 2: } \text{Alternate angles are equal}

    The two steps use different facts. Naming the first one for the second step is the commonest error here.

  12. Check the fact against the figure once more

    Alternate angles are equal\text{Alternate angles are equal}

    Angle QPB and angle PQC lie between the parallel lines on opposite sides of the transversal — the Z-shape — so they are alternate angles.

  13. Write the reason out in full

    Alternate angles are equal\text{Alternate angles are equal}

    A reason like "parallel lines" on its own earns nothing: the fact has to be named. Alternate angles are equal.

  14. Note the mark scheme

    1 mark: correct fact named\text{1 mark: correct fact named}

    One mark is available and it is for naming the correct angle fact.

  15. State the fact that earns the mark

    Alternate angles are equal\text{Alternate angles are equal}

    The reason that must be written down is: Alternate angles are equal.

Answer
Alternate angles are equal\text{Alternate angles are equal}
Question 3
5 markschallenging
ABAB and CDCD are parallel straight lines, with ABAB below CDCD. AA and CC are on the left; BB and DD are on the right. A straight line EFEF crosses ABAB at PP and crosses CDCD at QQ, with EE below ABAB and FF above CDCD. Angle APE=59∘APE = 59^\circ. Ben writes angle QPB=59∘QPB = 59^\circ, and then angle PQD=121∘PQD = 121^\circ. Which angle fact does Ben use in his second step?
Show worked solution

Worked solution

  1. Write down what you are given

    ∠QPB=59∘\angle QPB = 59^\circ

    Angle QPB is 59°. The question is which angle fact takes you from there to the angle that was written down.

  2. Look at the first step, which is not the one being asked about

    ∠QPB=59∘\angle QPB = 59^\circ

    The first step uses a different fact (vertically opposite angles are equal) to get angle QPB = 59°. The question asks about the SECOND step only.

  3. Describe where angle QPB and angle PQD sit

    ∠QPB=59∘,∠PQD=121∘\angle QPB = 59^\circ, \quad \angle PQD = 121^\circ

    Angle QPB and angle PQD lie between the parallel lines on the same side of the transversal — the C-shape — so they are co-interior angles.

  4. Name the fact

    Co-interior angles add up to 180∘\text{Co-interior angles add up to 180}^\circ

    Co-interior angles add up to 180°.

  5. Check the fact really does give that answer

    ∠PQD=180∘−59∘=121∘\angle PQD = 180^\circ - 59^\circ = 121^\circ

    Using the fact on angle QPB gives 121°, which is exactly what was written down.

  6. Check the numbers in the whole chain

    59∘→59∘→121∘59^\circ \rightarrow 59^\circ \rightarrow 121^\circ

    Both steps give the numbers that were written down, so the working is right — only the reason for the second step is in question.

  7. Rule out one of the other answers

    Vertically opposite angles are equal\text{Vertically opposite angles are equal}

    "Vertically opposite angles are equal." is a true fact, but it does not describe this pair: applying it here would give 59°, not 121°.

  8. Rule out one of the other answers

    Alternate angles are equal\text{Alternate angles are equal}

    "Alternate angles are equal." is a true fact, but it does not describe this pair: applying it here would give 59°, not 121°.

  9. Rule out one of the other answers

    Corresponding angles are equal\text{Corresponding angles are equal}

    "Corresponding angles are equal." is a true fact, but it does not describe this pair: applying it here would give 59°, not 121°.

  10. Rule out one of the other answers

    Co-interior angles are equal\text{Co-interior angles are equal}

    "Co-interior angles are equal." is simply not true, so it cannot justify anything. Used here it would give 59°.

  11. Compare the two steps

    step 1: Vertically opposite angles are equal,  step 2: Co-interior angles add up to 180∘\text{step 1: } \text{Vertically opposite angles are equal}, \; \text{step 2: } \text{Co-interior angles add up to 180}^\circ

    The two steps use different facts. Naming the first one for the second step is the commonest error here.

  12. Check the fact against the figure once more

    Co-interior angles add up to 180∘\text{Co-interior angles add up to 180}^\circ

    Angle QPB and angle PQD lie between the parallel lines on the same side of the transversal — the C-shape — so they are co-interior angles.

  13. Write the reason out in full

    Co-interior angles add up to 180∘\text{Co-interior angles add up to 180}^\circ

    A reason like "parallel lines" on its own earns nothing: the fact has to be named. Co-interior angles add up to 180°.

  14. Note the mark scheme

    1 mark: correct fact named\text{1 mark: correct fact named}

    One mark is available and it is for naming the correct angle fact.

  15. State the fact that earns the mark

    Co-interior angles add up to 180∘\text{Co-interior angles add up to 180}^\circ

    The reason that must be written down is: Co-interior angles add up to 180°.

Answer
Co-interior angles add up to 180∘\text{Co-interior angles add up to 180}^\circ
Question 4
6 markschallenging
ABAB and CDCD are parallel straight lines, with ABAB below CDCD. AA and CC are on the left; BB and DD are on the right. A straight line EFEF crosses ABAB at PP and crosses CDCD at QQ, with EE below ABAB and FF above CDCD. Angle QPB=(2x+13)∘QPB = (2x + 13)^\circ and angle DQF=(4x−17)∘DQF = (4x - 17)^\circ. Work out the size of angle PQDPQD.
Show worked solution

Worked solution

  1. Write down the two algebraic angles

    ∠QPB=(2x+13)∘,∠DQF=(4x−17)∘\angle QPB = (2x + 13)^\circ, \quad \angle DQF = (4x - 17)^\circ

    Both angles are written in terms of x. One angle fact will turn that into an equation.

  2. Name the fact that links angle QPB and angle DQF

    Corresponding angles are equal\text{Corresponding angles are equal}

    Corresponding angles are equal. Angle QPB and angle DQF sit in matching positions at the two crossings — the F-shape — so they are corresponding angles.

  3. Form an equation

    2x+13=4x−172x + 13 = 4x - 17

    Corresponding angles are equal. So the two expressions are equal, which gives an equation in x.

  4. Collect the like terms

    −2x=−30-2x = -30

    Gather the x terms on one side and the numbers on the other.

  5. Solve for x

    x=−30−2=15x = \frac{-30}{-2} = 15

    Dividing both sides by -2 gives x=15x = 15.

  6. Substitute x back to find angle QPB

    ∠QPB=2x+13=43∘\angle QPB = 2x + 13 = 43^\circ

    Putting x=15x = 15 into the first expression gives 43∘43^\circ.

  7. Substitute x back to find angle DQF

    ∠DQF=4x−17=43∘\angle DQF = 4x - 17 = 43^\circ

    Putting x=15x = 15 into the second expression gives 43∘43^\circ.

  8. Check the two angles against the fact

    43=4343 = 43

    Corresponding angles are equal. The two angles came out the same size, so x is right.

  9. Name the fact that links angle QPB and angle PQD

    Co-interior angles add up to 180∘\text{Co-interior angles add up to 180}^\circ

    Co-interior angles add up to 180°. Angle QPB and angle PQD lie between the parallel lines on the same side of the transversal — the C-shape — so they are co-interior angles.

  10. Work out angle PQD

    ∠PQD=180∘−43∘=137∘\angle PQD = 180^\circ - 43^\circ = 137^\circ

    Co-interior angles add up to 180°. Taking angle QPB, 43°, away from 180° leaves 137°.

  11. Check both angles are possible

    0∘<each angle<180∘0^\circ < \text{each angle} < 180^\circ

    Both angles came out between 0° and 180° (43° and 43°), so the figure can really be drawn.

  12. Avoid the usual mistake

    (2x+13)+(4x−17)=180 is wrong here(2x + 13) + (4x - 17) = 180 \text{ is wrong here}

    These angles are equal, not supplementary. Setting the two expressions to add up to 180 is the classic error and gives the wrong value of x.

  13. Write down the reasons that earn the marks

    corresponding angles are equal, then co-interior angles add up to 180°\text{corresponding angles are equal, then co-interior angles add up to 180°}

    In the exam the marks are for the reasons as much as for the numbers, so every step must name the angle fact it uses: corresponding angles are equal, then co-interior angles add up to 180°.

  14. Read the answer back into the figure

    ∠PQD=137∘\angle PQD = 137^\circ

    Put the numbers back on the diagram and check each angle looks the size it should — acute angles acute, obtuse angles obtuse.

  15. State the size of angle PQD

    ∠PQD=137∘\angle PQD = 137^\circ

    Angle PQD measures 137°.

Answer
137∘137^\circ
Question 5
5 markschallenging
ABAB and CDCD are parallel straight lines, with ABAB below CDCD. AA and CC are on the left; BB and DD are on the right. A straight line EFEF crosses ABAB at PP and crosses CDCD at QQ, with EE below ABAB and FF above CDCD. Angle QPB=(4x+7)∘QPB = (4x + 7)^\circ and angle PQC=(6x−13)∘PQC = (6x - 13)^\circ. Work out the size of angle BPEBPE.
Show worked solution

Worked solution

  1. Write down the two algebraic angles

    ∠QPB=(4x+7)∘,∠PQC=(6x−13)∘\angle QPB = (4x + 7)^\circ, \quad \angle PQC = (6x - 13)^\circ

    Both angles are written in terms of x. One angle fact will turn that into an equation.

  2. Name the fact that links angle QPB and angle PQC

    Alternate angles are equal\text{Alternate angles are equal}

    Alternate angles are equal. Angle QPB and angle PQC lie between the parallel lines on opposite sides of the transversal — the Z-shape — so they are alternate angles.

  3. Form an equation

    4x+7=6x−134x + 7 = 6x - 13

    Alternate angles are equal. So the two expressions are equal, which gives an equation in x.

  4. Collect the like terms

    −2x=−20-2x = -20

    Gather the x terms on one side and the numbers on the other.

  5. Solve for x

    x=−20−2=10x = \frac{-20}{-2} = 10

    Dividing both sides by -2 gives x=10x = 10.

  6. Substitute x back to find angle QPB

    ∠QPB=4x+7=47∘\angle QPB = 4x + 7 = 47^\circ

    Putting x=10x = 10 into the first expression gives 47∘47^\circ.

  7. Substitute x back to find angle PQC

    ∠PQC=6x−13=47∘\angle PQC = 6x - 13 = 47^\circ

    Putting x=10x = 10 into the second expression gives 47∘47^\circ.

  8. Check the two angles against the fact

    47=4747 = 47

    Alternate angles are equal. The two angles came out the same size, so x is right.

  9. Name the fact that links angle QPB and angle BPE

    Angles on a straight line add up to 180∘\text{Angles on a straight line add up to 180}^\circ

    Angles on a straight line add up to 180°. Angle QPB and angle BPE are next to each other on a straight line, so together they make a half turn.

  10. Work out angle BPE

    ∠BPE=180∘−47∘=133∘\angle BPE = 180^\circ - 47^\circ = 133^\circ

    Angles on a straight line add up to 180°. Taking angle QPB, 47°, away from 180° leaves 133°.

  11. Check both angles are possible

    0∘<each angle<180∘0^\circ < \text{each angle} < 180^\circ

    Both angles came out between 0° and 180° (47° and 47°), so the figure can really be drawn.

  12. Avoid the usual mistake

    (4x+7)+(6x−13)=180 is wrong here(4x + 7) + (6x - 13) = 180 \text{ is wrong here}

    These angles are equal, not supplementary. Setting the two expressions to add up to 180 is the classic error and gives the wrong value of x.

  13. Write down the reasons that earn the marks

    alternate angles are equal, then angles on a straight line add up to 180°\text{alternate angles are equal, then angles on a straight line add up to 180°}

    In the exam the marks are for the reasons as much as for the numbers, so every step must name the angle fact it uses: alternate angles are equal, then angles on a straight line add up to 180°.

  14. Read the answer back into the figure

    ∠BPE=133∘\angle BPE = 133^\circ

    Put the numbers back on the diagram and check each angle looks the size it should — acute angles acute, obtuse angles obtuse.

  15. State the size of angle BPE

    ∠BPE=133∘\angle BPE = 133^\circ

    Angle BPE measures 133°.

Answer
133∘133^\circ

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