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 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
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 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.

  2. Work out angle PQC

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

    Alternate angles are equal. Angle PQC is therefore the same size as angle QPB, which is 71°.

  3. State the size of angle PQC

    ∠PQC=71∘\angle PQC = 71^\circ

    Angle PQC measures 71°.

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 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.

  2. Work out angle PQD

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

    Co-interior angles add up to 180°. Taking angle QPB, 58°, away from 180° leaves 122°.

  3. State the size of angle PQD

    ∠PQD=122∘\angle PQD = 122^\circ

    Angle PQD measures 122°.

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 APQ and angle CQF

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

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

  2. Work out angle CQF

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

    Corresponding angles are equal. Angle CQF is therefore the same size as angle APQ, which is 115°.

  3. State the size of angle CQF

    ∠CQF=115∘\angle CQF = 115^\circ

    Angle CQF measures 115°.

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 APQ and angle PQD

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

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

  2. Work out angle PQD

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

    Alternate angles are equal. Angle PQD is therefore the same size as angle APQ, which is 124°.

  3. State the size of angle PQD

    ∠PQD=124∘\angle PQD = 124^\circ

    Angle PQD measures 124°.

Answer
124∘124^\circ

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