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Polygons

Polygons

Angles in parallel lines

Formulas Paper 1 + 2 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Corresponding, alternate, co-interior
  4. Ways they test it
  5. Mistakes to avoid
  6. Summary
  7. Practice
  8. Can I do this?
  9. Quiz

In this note

Syllabus 4.6: angles in parallel lines
  • 4.6Use corresponding angles (equal), alternate angles (equal) and co-interior angles (add to 180°).
  • 4.6Find angles in multi-step diagrams, with reasons.

Before you start: Angle facts. Next: Angles in polygons.

Formula bank

CorrespondingLearn itcorresponding angles are equalF-shape: same position at each crossing.Use it when parallel lines (arrows).
AlternateLearn italternate angles are equalZ-shape: opposite sides of the transversal, between the lines.Use it when parallel lines.
Co-interiorLearn itco-interior angles add up to 180°C-shape: same side, between the lines.Use it when parallel lines.

Corresponding, alternate, co-interior

Parallel lines are marked with matching arrows. A line crossing them (a transversal) meets each one at the same angle, which gives three rules. F, Z and C help you SPOT them, but the reason must use the proper name.

F-shape

Corresponding anglesaa
Same position at each crossing.

Corresponding angles are equal.

Z-shape

Alternate anglesaa
Opposite sides, between the lines.

Alternate angles are equal.

C-shape

Co-interior anglesab
Same side, between the lines.

Co-interior angles add up to 180°: \(a+b=180^\circ\).

General methodHow to find angles with parallel lines

  1. Find the parallel lines (arrows) and the transversal.
  2. Spot the shape linking a known angle to the unknown: F (corresponding), Z (alternate), C (co-interior).
  3. Write the value and the reason by name.
  4. A bend between the lines: draw an extra parallel line through the bend and use alternate angles twice.
Bent line between parallel lines35°48°35°48°
A bend: draw a third parallel line through it.

Top part 35° and bottom part 48° (both alternate angles), so the bend is 35° + 48° = 83°.

Triangle between parallel lines55°65°APQBC
Triangle ABC with A on line PQ, BC on the other parallel line.

Angle ABC = 55° and angle ACB = 65° (alternate angles), so angle BAC = 180 − 55 − 65 = 60°.

Try it yourself

1
Co-interior angles: one is 125°. The other?
Show answer

Answer 55

2
A bend between parallel lines makes 40° and 75° with them. The bend angle?
Show answer

Answer 115

Ways they test it

Type 1 Find a and b with reasons

"A transversal crosses two parallel lines... Find a and b, giving a reason for each." (4 marks)

Spot it parallel arrows and two letters.

  1. Name the pair for each.
  2. Value + reason.

For example transversal makes 64°

  1. alternate: 64°
  2. co-interior: 116°

Answer a = 64, b = 116

Type 2 Triangle between parallel lines

"PAQ is parallel to BC. Find angle BAC." (3 marks)

Spot it a triangle touching both lines.

  1. Alternate angles carry the angles into the triangle.
  2. Then angles in a triangle (or on the line PAQ).

Type 3 Algebra

"Co-interior angles are \((3x+10)^\circ\) and \((2x+20)^\circ\). Find x." (3 marks)

Spot it letters in two related angles.

  1. Equal or sum 180°?
  2. Form the equation and solve.

For example \(5x+30=180\)

    Answer x = 30

    Mistakes to avoid

    • ✗ Co-interior angles equal✓ add up to 180°Only alternate and corresponding are equal.
    • ✗ Using the rules without parallel lines✓ check for arrowsNo arrows, no rule.
    • ✗ 'Z angles' as a reason✓ 'alternate angles are equal'Mark scheme wording.

    Summary

    • Corresponding angles are equal (F).
    • Alternate angles are equal (Z).
    • Co-interior angles add up to 180° (C).
    • A bend: extra parallel line through it.

    If you see …

    • arrows on two lineslook for F, Z or C.
    • a bend between parallel linesadd the two alternate angles.

    Practice

    Practice: easy, medium, hard

    Easy

    1
    A transversal makes 72° with one of two parallel lines. The corresponding angle?
    Show answer

    Answer 72

    2
    The co-interior angle to 72°?
    Show answer

    Answer 108

    3
    Alternate angle to 64°?
    Show answer

    Answer 64

    Medium

    1
    A bend between parallel lines makes 35° with the top line and 48° with the bottom line. Find the bend angle.
    Guide me

    What is it asking? an angle

    What do we know? extra parallel line through the bend

    Plan 35 + 48

    Show working
    1. 83°

    Answer 83

    2
    PAQ is parallel to BC; angle PAB = 50°, angle QAC = 70°. Find angle BAC.
    Guide me

    What is it asking? an angle

    What do we know? alternate angles: 50° and 70° at B and C

    Plan 180 − 50 − 70

    Show working
    1. 60°

    Answer 60

    3
    PAQ is parallel to BC; angle PAB = 55°, angle QAC = 65°. Find angle ABC.
    Guide me

    What is it asking? an angle

    What do we know? alternate with angle PAB

    Plan equal

    Show working
    1. 55° (alternate angles)

    Answer 55

    Hard

    1
    Co-interior angles are \((3x+10)^\circ\) and \((2x+20)^\circ\). Find x.
    Guide me

    What is it asking? x

    What do we know? they add to 180

    Plan \(5x+30=180\)

    Show working
    1. \(5x=150\)
    2. \(x=30\)

    Answer 30

    PartAnswer / workingMarksGuidance
    \(5x+30=180\)M1
    30A1
    Total: 2 marks
    2
    Alternate angles are \((4x-15)^\circ\) and \((2x+25)^\circ\). Find the size of each angle.
    Guide me

    What is it asking? an angle

    What do we know? alternate angles are equal

    Plan \(4x-15=2x+25\)

    Show working
    1. \(2x=40\Rightarrow x=20\)
    2. \(4(20)-15=65\)

    Answer 65

    PartAnswer / workingMarksGuidance
    equationM1
    x = 20A1
    65°A1
    Total: 3 marks
    3
    A transversal crosses two parallel lines making 64° at the lower line. Find the co-interior angle at the upper line.
    Guide me

    What is it asking? an angle

    What do we know? co-interior

    Plan 180 − 64

    Show working
    1. 116°

    Answer 116

    PartAnswer / workingMarksGuidance
    116 with reasonB2
    Total: 2 marks

    Can I do this?

    Anything unticked: revise that section first. All ticked: take the quiz.

    Quiz

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