Mathematics • Year 8 • Unit 3 • Lesson 14

Parallel Lines in the Real World

Use F (corresponding), Z (alternate), and C (co-interior) angle rules where they actually appear: roads, railway tracks, ramps, ladder leaning against walls, and architectural plans.

Apply · Real-World Maths

1. Word problems

Each problem hides parallel lines and a transversal. Sketch it, label the angles, identify the F / Z / C shape, then solve. Every angle answer needs a reason — "65° (alternate angles, parallel lines)".

1.1 — Railway track and footbridge. Two railway tracks run parallel to each other. A pedestrian footbridge crosses both tracks at an angle. Where the bridge crosses the FIRST track, the bridge makes an angle of 70° with the track (measured on the south side, east of the bridge).

(a) What angle does the bridge make with the SECOND track on the south side, east of the bridge? Name the rule used.
(b) What angle does the bridge make with the second track on the south side, WEST of the bridge? Name the rule. 3 marks

Stuck? Same side, same position at each intersection = corresponding (F). Same side but on a straight line = supplementary.

1.2 — Roof rafters. The roof of a house has two parallel rafters supported by a vertical post. The angle between the LEFT rafter and the post is 55° (measured on the inside).

(a) The post acts as a transversal cutting both parallel rafters. Use co-interior angles to find the angle between the RIGHT rafter and the post on the inside of the roof.
(b) Give the reason. 3 marks

Stuck? Co-interior angles add to 180°. If one is 55°, the other = 180° − 55°.

1.3 — Wheelchair ramp. A wheelchair ramp leans against a vertical wall. The ground is horizontal. The ramp meets the ground at an angle of 12°. A safety rail runs parallel to the ramp.

(a) The safety rail also meets the (horizontal) ground at the same angle as the ramp. Explain why, using a parallel-line angle rule.
(b) The wall is perpendicular to the ground. What angle does the ramp make with the wall (where they meet)? Show your reasoning. 3 marks

Stuck? Two parallel lines (ramp and rail) cut by the ground → corresponding angles equal. Wall is perpendicular, so angle in the right triangle = 90° − 12°.

1.4 — Venetian blind slats. A venetian blind has 10 horizontal slats, all parallel to each other. A vertical string passes through all 10 slats. The angle between the string and the topmost slat is 78°.

(a) What angle does the string make with the 5th slat from the top? Give the reason.
(b) What angle does the string make with the 10th (bottom) slat? Give the reason. 3 marks

Stuck? All 10 slats are parallel. A line crossing parallel lines makes the same angle with EACH of them (corresponding angles equal).

1.5 — Soccer pitch lines. The two sidelines of a soccer pitch are parallel. The halfway line is perpendicular to the sidelines. A player runs diagonally across the pitch making an angle of 35° with one sideline.

(a) What angle does the player's run make with the OPPOSITE sideline (where it hits)? Give the reason.
(b) What angle does the player's run make with the halfway line? (Hint: the halfway line and the sideline are perpendicular, so use the right-triangle angle sum.) 3 marks

Stuck? Parallel sidelines cut by the player's path → alternate or corresponding angle to 35° = 35°. Half-line is perpendicular to sideline → use 90 − 35.

2. Explain your thinking

This question is about communication, not just answers. Use full sentences. 4 marks

2.1 A classmate has been asked to find the size of a co-interior angle when its partner is 73°. They write "73° again, because they are alternate angles". In your own words, explain (i) what mistake they have made, (ii) what the correct answer is and how to get it, and (iii) the difference between the Z-shape (alternate) and the C-shape (co-interior) in one or two clear sentences. Use the phrase "alternate are equal, co-interior add to 180°" somewhere in your answer.

Stuck? Revisit lesson § Card 6 — co-interior angles are on the SAME side of the transversal, and they ADD to 180°.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Railway track and footbridge

(a) 70° (corresponding angles, parallel lines — same position at each intersection).
(b) 180° − 70° = 110° (angles on a straight line; alternatively co-interior with original 70°).

1.2 — Roof rafters

(a) Right rafter angle with post (inside) = 180° − 55° = 125°.
(b) Reason: co-interior angles add to 180° (parallel lines).

1.3 — Wheelchair ramp

(a) The ramp and the safety rail are parallel; the ground is a transversal. Corresponding angles (parallel lines) are equal, so the rail also meets the ground at 12°.
(b) The ramp, ground and wall form a right triangle. Angle ramp–wall = 90° − 12° = 78° (angle sum in a right triangle).

1.4 — Venetian blind

(a) 78° (corresponding angles, parallel lines).
(b) 78° (corresponding angles, parallel lines — applies to any of the 10 parallel slats).

1.5 — Soccer pitch

(a) 35° (alternate angles, parallel lines — or corresponding, depending on which 35° we measure).
(b) Halfway line is perpendicular to the sideline. The player's run makes 35° with the sideline, so it makes 90° − 35° = 55° with the halfway line.

2.1 — Explain your thinking (sample response)

The classmate has confused two different rules. Co-interior angles look similar to alternate angles, but they sit on the SAME side of the transversal (forming a C-shape), not opposite sides (Z-shape). The rule is: alternate are equal, co-interior add to 180°. So the correct co-interior partner of 73° is 180° − 73° = 107°, NOT 73°. To tell the two apart: look at where the two angles sit relative to the transversal — opposite sides = Z = equal; same side = C = supplementary.

Marking: 1 mark for spotting "used wrong rule"; 1 mark for the correct answer of 107°; 1 mark for the Z vs C distinction; 1 mark for using "alternate are equal, co-interior add to 180°".