GCSE Angles in parallel lines Practice Questions

Free GCSE Angles in parallel lines practice questions with full step-by-step worked solutions. Covers corresponding angles, parallel lines, alternate angles, co-interior angles. Practise exam-style problems and check your method.

corresponding anglesparallel linesalternate anglesco-interior anglesvertically opposite anglesangles on a straight line
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
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=62∘QPB = 62^\circ. Work out the size of angle DQFDQF.
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Worked solution

  1. 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.

  2. Work out angle DQF

    ∠DQF=∠QPB=62∘\angle DQF = \angle QPB = 62^\circ

    Corresponding angles are equal. Angle DQF is therefore the same size as angle QPB, which is 62°.

  3. State the size of angle DQF

    ∠DQF=62∘\angle DQF = 62^\circ

    Angle DQF measures 62°.

Answer
62∘62^\circ
Question 2
2 markseasy
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=124∘QPB = 124^\circ. Mia says that angle PQD=56∘PQD = 56^\circ. Which angle fact shows that Mia is right?
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Worked solution

  1. Describe where angle QPB and angle PQD sit

    ∠QPB=124∘,∠PQD=56∘\angle QPB = 124^\circ, \quad \angle PQD = 56^\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.

  2. 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°.

  3. Rule out the other answers

    the other four do not fit\text{the other four do not fit}

    "Alternate angles are equal." is a true fact, but it does not describe this pair: applying it here would give 124°, not 56°. "Corresponding angles are equal." is a true fact, but it does not describe this pair: applying it here would give 124°, not 56°. "Co-interior angles are equal." is simply not true, so it cannot justify anything. Used here it would give 124°. "Co-interior angles add up to 360°." is simply not true, so it cannot justify anything. Used here it would give 236°.

Answer
Co-interior angles add up to 180∘\text{Co-interior angles add up to 180}^\circ
Question 3
2 marksintermediate
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 PQC=133∘PQC = 133^\circ. Ada says that angle PQD=47∘PQD = 47^\circ. Which angle fact shows that Ada is right?
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Worked solution

  1. Write down what you are given

    ∠PQC=133∘\angle PQC = 133^\circ

    Angle PQC is 133°. The question is which angle fact takes you from there to the angle that was written down.

  2. Describe where angle PQC and angle PQD sit

    ∠PQC=133∘,∠PQD=47∘\angle PQC = 133^\circ, \quad \angle PQD = 47^\circ

    Angle PQC and angle PQD are next to each other on a straight line, so together they make a half turn.

  3. Name the fact

    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°.

  4. Check the fact really does give that answer

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

    Using the fact on angle PQC gives 47°, which is exactly what was written down.

  5. Rule out the other answers

    the other four do not fit\text{the other four do not fit}

    "Co-interior angles add up to 180°." is a true fact and it happens to give the same number, but it is the wrong reason — angle PQC and angle PQD are not that kind of pair, so it would earn no mark. "Vertically opposite angles are equal." is a true fact, but it does not describe this pair: applying it here would give 133°, not 47°. "Angles on a straight line add up to 360°." is simply not true, so it cannot justify anything. Used here it would give 227°. "Corresponding angles are equal." is a true fact, but it does not describe this pair: applying it here would give 133°, not 47°.

  6. State the fact that earns the mark

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

    The reason that must be written down is: Angles on a straight line add up to 180°.

Answer
Angles on a straight line add up to 180∘\text{Angles on a straight line add up to 180}^\circ
Question 4
3 markshard
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 BPE=52∘BPE = 52^\circ. Iris says that angle PQD=52∘PQD = 52^\circ. Which angle fact shows that Iris is right?
Show worked solution

Worked solution

  1. Write down what you are given

    ∠BPE=52∘\angle BPE = 52^\circ

    Angle BPE is 52°. The question is which angle fact takes you from there to the angle that was written down.

  2. Describe where angle BPE and angle PQD sit

    ∠BPE=52∘,∠PQD=52∘\angle BPE = 52^\circ, \quad \angle PQD = 52^\circ

    Angle BPE and angle PQD sit in matching positions at the two crossings — the F-shape — so they are corresponding angles.

  3. Name the fact

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

    Corresponding angles are equal.

  4. Check the fact really does give that answer

    ∠PQD=∠BPE=52∘\angle PQD = \angle BPE = 52^\circ

    Using the fact on angle BPE gives 52°, which is exactly what was written down.

  5. 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 128°, not 52°.

  6. 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 BPE and angle PQD are not that kind of pair, so it would earn no mark.

  7. 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 128°.

  8. 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 128°, not 52°.

  9. Note what the mark is for

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

    The number was already given in the question. The mark here is entirely for naming the angle fact that justifies it, so a vague answer such as "because the lines are parallel" would score nothing.

  10. State the fact that earns the mark

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

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

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

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