Mathematics • Year 7 • Unit 3 • Lesson 11
Parallel Lines — Real World
Railway tracks, ladders against walls, pedestrian crossings, garden trellis, roof beams — parallel lines and transversals show up everywhere. Use the Z, C and F rules to work out unknown angles in genuine situations.
1. Word problems
For each problem: (i) name the angle relationship (alt. / co-int. / corr.) and (ii) work out the unknown. Cite the parallel lines.
1.1 — Railway tracks. Two railway tracks run perfectly parallel. A footbridge crosses both at an angle. On the LEFT track, the bridge makes an angle of 68° measured above the track on the right side of the bridge. On the RIGHT track, what is the angle measured above the track on the right side of the bridge?
(a) Which angle family is this? (b) Find the angle. 2 marks
1.2 — Pedestrian crossing. Two white painted lines on a road mark the edges of a pedestrian crossing — they're parallel. A cyclist rides diagonally across both lines. The cyclist's path makes an angle of 110° with the first line, measured inside the crossing on the right of the cyclist's path. The angle the cyclist's path makes with the second line, measured INSIDE the crossing on the LEFT of the cyclist's path, is what?
(a) Which angle family? (b) Find the angle. 2 marks
1.3 — Garden trellis. A wooden trellis has two horizontal rails that are parallel. A diagonal slat is nailed across both rails. On the TOP rail, the slat makes an angle of 125° measured below the rail on the right of the slat. On the BOTTOM rail, what is the angle measured ABOVE the rail on the right of the slat?
(a) Which angle family connects these two angles? (b) Find the angle. 2 marks
1.4 — Roof beams. Two horizontal roof beams in a barn are parallel. A diagonal support brace runs from the top beam down to the bottom beam. The brace makes a 38° angle with the top beam (measured below the top beam, on the LEFT of the brace). The brace also makes an angle with the bottom beam (measured above the bottom beam, on the RIGHT of the brace).
(a) Which angle family? (b) Find that angle. 2 marks
1.5 — Tennis court lines. The two singles sidelines on a tennis court are parallel. A diagonal chalk line is drawn across both. At the LEFT sideline, two angles are formed: one is 152° (the larger one). At the RIGHT sideline, find: (a) the corresponding angle to 152°, (b) the alternate angle to 152°, and (c) a co-interior angle to 152°.
Show your reasoning for each. 3 marks
2. Explain your thinking
Full sentences. 4 marks
2.1 A carpenter is checking that two shelves are parallel. She places a long straight edge across both shelves as a transversal. She measures two corresponding angles formed at each shelf: the angle at the top shelf is 89° and the angle at the bottom shelf is 91°. In a short paragraph: (i) Are the shelves parallel? (ii) Explain how you know, using the corresponding-angles rule. (iii) What should the bottom angle be for the shelves to be exactly parallel?
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Railway tracks
(a) Same position at each track → corresponding angles (F).
(b) Equal: the right-track angle is 68° (corr. ∠s, left track ∥ right track).
1.2 — Pedestrian crossing
(a) Both inside the crossing, opposite sides of the cyclist's path → alternate angles (Z).
(b) Equal: angle = 110° (alt. ∠s, line 1 ∥ line 2).
1.3 — Garden trellis
The 125° is below the top rail on the right of the slat = inside the C, on the right side. The angle above the bottom rail on the right of the slat = inside the C, on the same side. So they form a C → co-interior.
(b) Supplementary: angle = 180 − 125 = 55° (co-int. ∠s, top rail ∥ bottom rail).
1.4 — Roof beams
(a) Both between the beams, on opposite sides of the brace → alternate (Z).
(b) Equal: angle = 38° (alt. ∠s, top beam ∥ bottom beam).
1.5 — Tennis court lines (152°)
(a) Corresponding to 152° on the right sideline: same position at the other crossing → equal: 152° (corr. ∠s, left ∥ right).
(b) Alternate to 152°: opposite side of transversal, between the lines → equal: 152° (alt. ∠s, left ∥ right).
(c) Co-interior to 152°: same side of transversal, between the lines → supplementary: 180 − 152 = 28° (co-int. ∠s, left ∥ right).
2.1 — Shelves and the straight edge (sample response)
If two shelves are truly parallel, then the corresponding angles formed by a transversal cutting both must be EQUAL (from Lesson 11, the corresponding-angles / F-shape rule). The carpenter measured 89° on the top shelf and 91° on the bottom shelf. Because 89° ≠ 91°, the shelves are not exactly parallel. For the shelves to be parallel, the bottom angle must also be 89° — matching the top — so the F-shape gives two equal corresponding angles.
Marking: 1 for "not parallel"; 1 for citing corresponding-angles rule; 1 for explaining the 89 ≠ 91 contradiction; 1 for the corrected 89°.