Mathematics • Year 10 • Unit 3 • Lesson 9
Parallel Lines in the Built World
Apply Lesson 9's three rules (corresponding equal, alternate equal, co-interior supplementary) to real settings: railway tracks crossed by a level-crossing road, parking-bay markings on a driveway, sloped seating in a stadium, and parallel rulers on a navigation chart. Always state the reason — examiners take half a mark per missing reason.
1. Word problems
For each problem: identify which rule applies (corresponding, alternate, or co-interior), state it in brackets, then calculate. A correct number with no reason only earns half marks.
1.1 — Railway level crossing. Two parallel railway tracks are crossed by a country road that runs at an angle. At the upper track, the acute angle between the road and the rail is 38°.
(a) Find the corresponding acute angle at the lower track.
(b) Find the obtuse angle on the same side of the road at the upper track. 3 marks
1.2 — Diagonal parking bays. A row of parking bays is marked with parallel white lines crossed by a single diagonal driving lane. The driving lane meets one parking line at 65°.
(a) Find the alternate angle on the next parking line (the "Z" pair).
(b) Find the co-interior angle to the 65° angle. 3 marks
1.3 — Stadium seating. Two parallel rows of stadium seats are connected by a sloping staircase that crosses each row. The acute angle between the staircase and the upper row is 33°.
(a) Find the acute angle between the staircase and the lower row, with a reason.
(b) A safety officer claims that for the rows to be exactly parallel the alternate angles must be equal. Use your answer to (a) to confirm or reject this claim. 3 marks
1.4 — Navigation chart. Two parallel rulers are used to transfer a heading across a marine chart. The first ruler crosses a line of latitude at 142° (measured on the upper side).
(a) What is the corresponding angle where the second (parallel) ruler crosses a line of latitude further down the chart?
(b) What is the acute angle on the same intersection? 3 marks
1.5 — Floorboards across a hallway. Long floorboards run parallel along a hallway. A diagonal strip of skirting board is laid across them at an angle of 117° to one of the floorboards (measured between the skirting and the board, on the upper side).
(a) Find the co-interior angle on the next floorboard.
(b) Find the alternate angle on the next floorboard. 3 marks
2. Explain your thinking
This question is about reasoning, not just numbers. Use full sentences. 4 marks
2.1 A friend says: "Corresponding angles are always equal — even if the two lines aren't parallel — because they're 'in matching positions'." Using Lesson 9, explain (i) what part of the friend's statement is reasonable, (ii) why the claim is mathematically wrong, (iii) the rule that does hold whether or not lines are parallel (give a short example), and (iv) name the geometric concept "lines are parallel ⇔ corresponding angles equal" and explain how it lets us prove two lines are parallel. Use the word "converse" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Railway level crossing
(a) Corresponding angle on the lower track = 38° (corresponding angles on parallel lines are equal).
(b) Obtuse angle on the same side at the upper track = 180° − 38° = 142° (angles on a straight line are supplementary).
1.2 — Diagonal parking bays
(a) Alternate angle on the next parking line = 65° (alternate angles on parallel lines are equal — Z pattern).
(b) Co-interior angle = 180° − 65° = 115° (co-interior angles on parallel lines are supplementary — C pattern).
1.3 — Stadium seating
(a) Acute angle on the lower row = 33° (alternate angles on parallel lines are equal).
(b) Confirmed. Because the alternate angles are both 33° (equal), the converse of the alternate-angle theorem says the rows must be parallel. The safety officer's claim is correct.
1.4 — Navigation chart
(a) Corresponding angle on the second ruler crossing = 142° (corresponding angles on parallel lines are equal — F pattern).
(b) Acute angle on the same intersection = 180° − 142° = 38° (angles on a straight line are supplementary).
1.5 — Floorboards and skirting strip
(a) Co-interior angle on the next floorboard = 180° − 117° = 63° (co-interior angles on parallel lines are supplementary).
(b) Alternate angle on the next floorboard = 117° (alternate angles on parallel lines are equal).
2.1 — Explain your thinking (sample response)
(i) The friend is on the right track in noticing that corresponding angles sit in matching positions at each intersection. That part of the language is fine. (ii) But the claim that they are "always equal" is wrong: corresponding angles are equal only when the two lines are parallel. If the lines diverge, the matching positions still exist but the angles will differ — they would only be equal if the transversal hit both lines at the same angle, which is exactly what "parallel" means. (iii) A rule that does hold without needing parallel lines is vertically opposite angles are equal — for example, two streets crossing at a roundabout always produce two pairs of equal vertically-opposite angles regardless of any other geometry. (iv) The biconditional "lines parallel ⇔ corresponding angles equal" gives us the converse theorem: if we measure two corresponding angles and find them equal, we can conclude the lines must be parallel. This is how surveyors verify parallelism without an infinite ruler.
Marking: 1 for naming what is reasonable (matching positions), 1 for the "only when parallel" correction, 1 for vertically opposite angles as an example, 1 for using "converse" correctly and explaining the proof-of-parallel application.