Practice Maths

L58 — Angles and Parallel Lines

Key Terms

parallel lines
Lines that never meet. Marked with matching arrows (>>) in diagrams. Always the same distance apart.
transversal
A line that crosses two or more parallel lines. Creates 8 angles — 4 at each intersection.
vertically opposite angles
Angles directly across from each other at the same intersection. Always equal.
supplementary angles
Two angles that add to 180°. Adjacent angles on a straight line are always supplementary.

The 8 Angles Formed

When a transversal cuts two parallel lines, 4 angles form at each intersection. From one angle you can find all others using two rules:

  • Vertically opposite: the angle directly across = equal
  • Adjacent (on a straight line): the angle beside = 180° − first angle
a b c d e f g h m n t

Purple angles (a, d, e, h) are one size  •  Red angles (b, c, f, g) are the other size

Hot Tip: Know one angle → you know all 8. The two sizes always add to 180°.

Worked Example

One angle at an intersection is 65°. Find all other 7 angles.

Step 1 — Vertically opposite: angle directly across = 65°.

Step 2 — Straight line: angles beside = 180° − 65° = 115°. Both of those are vertically opposite to each other too.

Step 3 — Other intersection: identical pattern → 65°, 65°, 115°, 115°.

Answer: Four angles of 65° and four angles of 115°.

a b c d e f g h

Blue lines are parallel (>>). Red line is the transversal.

What Are Parallel Lines?

Parallel lines run in the same direction and are always the same distance apart. They never meet. In diagrams they are marked with matching arrows (>>).

Examples: railway tracks, edges of a ruler, rows of seats in a cinema.

What Is a Transversal?

A transversal crosses two or more lines. When it crosses two parallel lines it creates 8 angles — 4 at each intersection. Knowing just one angle lets you find all the rest.

The Two Key Rules

  • Vertically opposite angles: When two lines cross, the angles directly across from each other are always equal. E.g. if one angle is 65°, the vertically opposite angle is also 65°.
  • Angles on a straight line: Angles that sit on a straight line always add to 180°. So if one angle is 65°, the next one around the intersection is 115°.

Finding All 8 Angles

Because the parallel lines are identical, both intersections have exactly the same four angles. So there are really only two different angle sizes: the original angle and its supplement (180° minus the original).

Key tip: Always check for the >> arrows before using parallel-line rules. If no arrows are shown, the lines may not be parallel and the rules don’t apply.

  1. Angles on a Straight Line Fluency

    Find the value of x in each diagram. Angles on a straight line add to 180°.

    65° x°
    (a)
    130° x°
    (b)
    48° x°
    (c)
    x° 72°
    (d)

  2. Vertically Opposite Angles Fluency

    Find the value of x. Vertically opposite angles are equal.

    40° x°
    (a)
    115° x°
    (b)
    62° x°
    (c)
    x° 52°
    (d)

  3. Find x at an Intersection Fluency

    In each diagram, find x. Use vertically opposite angles or angles on a straight line.

    74° x°
    (a) find x (on a straight line)
    55° x°
    (b) find x (vertically opposite or straight line?)
    88° x°
    (c) find x
    x° 44°
    (d) find x

  4. All Eight Angles Understanding

    Two parallel lines are cut by a transversal. One angle is marked. Use vertically opposite and straight-line rules to find all other 7 angles. Label each on a sketch.

    52°
    (a) Find all 7 other angles.
    118°
    (b) Find all 7 other angles.
    73°
    (c) Find all 7 other angles.
    90°
    (d) Find all 7 other angles.

  5. Algebra on a Straight Line Understanding

    Find the value of x in each diagram. Then state both angle sizes.

    (2x+10)° (3x−20)°
    (a)
    3x° x° 2x°
    (b) Three angles on a straight line
    (5x−8)° (3x+16)°
    (c) Vertically opposite — find x
    (4x+5)° (6x−25)°
    (d)

  6. Name the Angle Relationship Understanding

    For each pair of marked angles, state whether they are vertically opposite, supplementary (on a straight line), or neither. Then find x.

    58° x°
    (a)
    37° x°
    (b)
    49° x°
    (c)
    x° 121°
    (d)

  7. Are These Angle Values Possible? Understanding

    Each diagram shows angle values at an intersection. Decide whether the values are correct. Give a reason for each answer.

    95° 85°?
    (a) Is 85° correct for the blue angle?
    60° 60° 60°
    (b) Can three 60° angles fit on a straight line?
    70° 110° 70° 110°
    (c) Are these four angle values consistent?
    90° 90°
    (d) How many distinct angle sizes exist across all 8 angles?

  8. Real-World Parallel Lines Problem Solving

    52° beam 1 beam 2 brace
    (a) A roof brace crosses two parallel beams at 52°. List all 8 angles the brace makes.
    68° rail 1 rail 2 road
    (b) A road crosses two parallel railway lines. The acute angle at the first crossing is 68°. Find all 8 angles, giving a reason for each.

  9. Algebra and Reasoning Problem Solving

    x° (180−x)° x° (180−x)°
    (a) Write expressions for all 4 angles. Show they add to 360° for any x.
    (3n)° (180−3n)°
    (b) Show algebraically these two angles always add to 180° for any n.

    (c) At a point, five angles are formed: 70°, 50°, x°, 80°, and y°, where x and y are vertically opposite. All five angles add to 360°. Find x and y.

  10. Investigation Problem Solving

    60° 40° ?°
    Two transversals cross two parallel lines and meet between them.
    1. How many distinct angle sizes appear altogether across both intersections on each parallel line? List them.
    2. What is the angle where the two transversals meet (marked ?° above)? Hint: angles of a triangle add to 180°.
    3. The transversals meet at angle t°, one makes p° with the parallel line. Write a formula for angle q° that the other transversal makes, in terms of t and p.
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