Mathematics • Year 7 • Unit 3 • Lesson 12

Parallel Line Properties — Real World

Apply the alternate/co-interior/corresponding rules to bookshelves, basketball court lines, road signs, gates and architecture. Then turn the rules around to prove whether sides of a real object are parallel.

Apply · Real-World Maths

1. Word problems

For each problem: (i) identify the rule, (ii) write the equation with reason, (iii) state the answer.

1.1 — Bookshelves. Two bookshelves are mounted on a wall and are parallel to each other. A diagonal cable is run from the top-left of the upper shelf to the bottom-right of the lower shelf. The cable makes a 35° angle with the upper shelf, measured below the shelf on the right of the cable. What angle does the cable make with the lower shelf, measured above the shelf on the LEFT of the cable?

2 marks

Stuck? Both angles are between the shelves, on opposite sides of the cable. Z-shape → alternate, equal.

1.2 — Basketball court. The two free-throw lines on a basketball court are parallel. A diagonal pass crosses both lines. The angle the pass makes with the FAR free-throw line, measured below the line on the right of the pass, is 142°. Find the angle the pass makes with the NEAR free-throw line, measured ABOVE the line on the RIGHT of the pass.

2 marks

Stuck? Same side of the pass, between the lines = co-interior, sum 180°.

1.3 — Gate. A garden gate has two horizontal cross-bars that are parallel. A diagonal brace runs from the top-left of the gate to the bottom-right. On the top bar, the brace makes an angle of (3a)° above the bar on the RIGHT of the brace. On the bottom bar, the brace makes an angle of (a + 40)° above the bar on the RIGHT of the brace. The bars are parallel.

(a) Identify the angle family. (b) Form an equation, solve for a. (c) State the angle. 3 marks

Stuck? Both above the bar, both on the right of the brace = same position = corresponding, equal.

1.4 — Architecture. An architect designs a building with two long parallel facade lines (top of building and ground line, both horizontal). A diagonal stripe crosses both. The stripe makes a 67° angle with the top line above the line on the LEFT of the stripe. The shadow of the stripe on the ground falls between the two facade lines and makes an angle with the ground above the line on the RIGHT of the stripe. Find the shadow's angle.

2 marks

Stuck? Above the top line on the left, above the ground line on the right — opposite sides of the stripe, between the lines isn't quite right because the top angle is outside. Use straight-line first to get the angle below the top line on the left (180 − 67), then alternate to the ground.

1.5 — Prove parallel. A builder is checking that two roof rafters are parallel. He uses a long ruler as a transversal and measures the alternate angles formed at each rafter. He gets 47° at the upper rafter and 47° at the lower rafter.

(a) Are the rafters parallel? (b) Cite the converse rule that justifies your answer. 2 marks

Stuck? Equal alternate angles → lines parallel (converse of the alt. ∠s rule).

2. Explain your thinking

Full sentences. 4 marks

2.1 A student writes the following reason in her work: "x = 64° (alt. angles)". Her teacher takes off marks. In a short paragraph, explain (i) what is MISSING from the student's reason, (ii) what the FULL Year 7 reason phrase looks like, and (iii) why citing the parallel lines matters. Refer to the lesson's "reason phrase" requirement.

Stuck? Revisit lesson § "Reason Phrases You Must Write" — the parallel lines name belongs in the reason.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Bookshelves (35°)

Between the shelves, opposite sides of the cable → alternate (Z), equal.
Angle on lower shelf = 35° (alt. ∠s, upper shelf ∥ lower shelf).

1.2 — Basketball court (142°)

Between the two lines, same side of the pass → co-interior (C), supplementary.
Angle = 180 − 142 = 38° (co-int. ∠s, far line ∥ near line).

1.3 — Gate (algebra)

(a) Same position at each bar → corresponding angles (equal).
(b) 3a = a + 40 (corr. ∠s, top bar ∥ bottom bar)
2a = 40 → a = 20.
(c) Angle = 3(20)° = 60°. Check: a + 40 = 20 + 40 = 60° ✓.

1.4 — Architecture (67°)

Step 1 — the 67° is OUTSIDE the parallel lines (above the top). Use straight-line to get its partner BELOW the top line on the left: 180 − 67 = 113° (∠s on a straight line).
Step 2 — That 113° (between the lines, on the left) is alternate with the shadow angle (between the lines, on the right) → equal.
Shadow angle = 113° (alt. ∠s, top line ∥ ground line).
Alternative: corresponding (67°) to ground line above-left, then straight-line to above-right: 180 − 67 = 113°. Same answer.

1.5 — Prove parallel rafters

(a) Yes, the rafters are parallel.
(b) The two alternate angles at the transversal are equal (47° = 47°). By the converse of the alternate-angles rule, equal alternate angles guarantee the lines are parallel. ∴ upper rafter ∥ lower rafter.

2.1 — Reason phrase critique (sample response)

The student's reason "x = 64° (alt. angles)" is incomplete because it does NOT name the parallel lines. The full Year 7 reason phrase is "(alt. ∠s, AB ∥ CD)" — the rule name PLUS the parallel-line names. Citing the parallel lines matters because the alternate-angles rule ONLY works when the two lines being cut by the transversal are parallel; if they aren't parallel, the angles aren't guaranteed to be equal. By writing the parallel-line names in the reason, the student proves she has checked that condition.

Marking: 1 for "missing parallel-line names"; 1 for the full reason format; 1 for "rule only works when parallel"; 1 for clear full-sentence explanation.