Angles in parallel lines Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Angles in parallel lines questions. See exactly how to solve problems on corresponding angles, parallel lines, alternate angles, co-interior angles.

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.

Worked solution

  1. Name the fact that links angle QPBQPB and angle DQFDQF

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

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

  2. Work out angle DQFDQF

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

    Corresponding angles are equal. Angle DQFDQF is therefore the same size as angle QPBQPB, which is 62∘62^{\circ}.

  3. State the size of angle DQFDQF

    ∠DQF=62∘\angle DQF = 62^\circ

    Angle DQFDQF measures 62∘62^{\circ}.

Answer
62∘62^\circ
Question 2
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=71∘QPB = 71^\circ. Work out the size of angle PQCPQC.

Worked solution

  1. Name the fact that links angle QPBQPB and angle PQCPQC

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

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

  2. Work out angle PQCPQC

    ∠PQC=∠QPB=71∘\angle PQC = \angle QPB = 71^\circ

    Alternate angles are equal. Angle PQCPQC is therefore the same size as angle QPBQPB, which is 71∘71^{\circ}.

  3. State the size of angle PQCPQC

    ∠PQC=71∘\angle PQC = 71^\circ

    Angle PQCPQC measures 71∘71^{\circ}.

Answer
71∘71^\circ
Question 3
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=58∘QPB = 58^\circ. Work out the size of angle PQDPQD.

Worked solution

  1. Name the fact that links angle QPBQPB and angle PQDPQD

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

    Co-interior angles add up to 180∘180^{\circ}. Angle QPBQPB and angle PQDPQD lie between the parallel lines on the same side of the transversal — the C-shape — so they are co-interior angles.

  2. Work out angle PQDPQD

    ∠PQD=180∘−58∘=122∘\angle PQD = 180^\circ - 58^\circ = 122^\circ

    Co-interior angles add up to 180∘180^{\circ}. Taking angle QPBQPB, 58∘58^{\circ}, away from 180∘180^{\circ} leaves 122∘122^{\circ}.

  3. State the size of angle PQDPQD

    ∠PQD=122∘\angle PQD = 122^\circ

    Angle PQDPQD measures 122∘122^{\circ}.

Answer
122∘122^\circ
Question 4
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 APQ=115∘APQ = 115^\circ. Work out the size of angle CQFCQF.

Worked solution

  1. Name the fact that links angle APQAPQ and angle CQFCQF

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

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

  2. Work out angle CQFCQF

    ∠CQF=∠APQ=115∘\angle CQF = \angle APQ = 115^\circ

    Corresponding angles are equal. Angle CQFCQF is therefore the same size as angle APQAPQ, which is 115∘115^{\circ}.

  3. State the size of angle CQFCQF

    ∠CQF=115∘\angle CQF = 115^\circ

    Angle CQFCQF measures 115∘115^{\circ}.

Answer
115∘115^\circ
Question 5
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 APQ=124∘APQ = 124^\circ. Work out the size of angle PQDPQD.

Worked solution

  1. Name the fact that links angle APQAPQ and angle PQDPQD

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

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

  2. Work out angle PQDPQD

    ∠PQD=∠APQ=124∘\angle PQD = \angle APQ = 124^\circ

    Alternate angles are equal. Angle PQDPQD is therefore the same size as angle APQAPQ, which is 124∘124^{\circ}.

  3. State the size of angle PQDPQD

    ∠PQD=124∘\angle PQD = 124^\circ

    Angle PQDPQD measures 124∘124^{\circ}.

Answer
124∘124^\circ

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