Mathematics • Year 7 • Unit 3 • Lesson 1
Angles — Real World
Use complementary, supplementary, vertically opposite and angles-at-a-point relationships to solve practical problems — skateboard turns, roof pitches, scissors, clock hands and signpost layouts. State a reason for every answer.
1. Word problems
Set up the relationship, solve it, and state a geometric reason. A bare number without a reason only earns half marks.
1.1 — Skateboard turn. A skater performs a 90° turn (a "shove-it"), then immediately spins another 145°. What total angle have they turned through? Classify that total. 2 marks
1.2 — Roof pitch. A pitched roof meets the wall so that the angle between the rafter and the wall is 35°. The wall is vertical (the rafter and the wall form a corner — a right angle would be 90°). What is the angle between the rafter and the horizontal ceiling joist?
(a) Identify whether the two angles are complementary or supplementary.
(b) Find the missing angle. State the reason. 2 marks
1.3 — Open scissors. The two blades of a pair of scissors cross at a single point. The angle between the upper blade and the lower handle (on the right side) is 38°. Find the three other angles formed at the cross. Justify each with a reason. 3 marks
1.4 — Clock hands at 4 o'clock. At exactly 4 o'clock, the hour hand points to 4 and the minute hand points to 12. The angle between them (the smaller one) is the angle you'd see on the clock face. A clock face is split into 12 equal sectors meeting at the centre. (a) How many degrees is each sector? (b) What is the smaller angle between the hands at 4:00? (c) Classify it. 3 marks
1.5 — Five-way roundabout. A roundabout has five roads meeting at one centre point. Four of the angles between consecutive roads (going around) are 70°, 65°, 80° and 75°. (a) What is the fifth angle? (b) State the reason. 2 marks
2. Explain your thinking
Use full sentences. 4 marks
2.1 A classmate writes "If two angles sit next to each other on a straight line, they must be equal because they share a side." Explain in your own words (i) why that statement is wrong, (ii) what the correct relationship between the two angles actually is, and (iii) under what specific condition the two angles WOULD be equal. Use the words supplementary and vertically opposite somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Skateboard turn
Total = 90 + 145 = 235°. Classification: 180° < 235° < 360° → reflex.
1.2 — Roof pitch
(a) The rafter and the horizontal joist form the 90° corner at the wall, so the two angles are complementary.
(b) 90 − 35 = 55° (comp. ∠s).
1.3 — Open scissors
Vertically opposite the 38° = 38° (vert. opp. ∠s).
Adjacent along the line = 180 − 38 = 142° (∠s on str. line).
Fourth angle = 142° (vert. opp. ∠s).
Check: 38 + 142 + 38 + 142 = 360° ✓.
1.4 — Clock at 4:00
(a) Each sector = 360 ÷ 12 = 30° (∠s at a pt.).
(b) From 12 to 4 covers 4 sectors → 4 × 30 = 120°.
(c) 90° < 120° < 180° → obtuse.
1.5 — Five-way roundabout
(a) Sum of four known angles = 70 + 65 + 80 + 75 = 290. Fifth angle = 360 − 290 = 70°.
(b) Reason: ∠s at a pt. sum to 360°.
2.1 — Explain your thinking (sample response)
The classmate is wrong because two angles sitting next to each other on a straight line are supplementary, not equal. Sharing a side doesn't make them equal — it means their two outer arms together form a straight line, so the two angles must add to 180°. For example, if one angle is 110°, the other must be 70°. The two angles WOULD be equal only when each is exactly 90° (so 90° + 90° = 180°), which happens when the dividing ray is perpendicular to the straight line. The classmate may be confusing this with vertically opposite angles, which ARE equal — but those are angles across the X of two crossing lines, not side-by-side angles on one line.
Marking: 1 for naming supplementary; 1 for the correct relationship "sum 180°"; 1 for the specific 90° + 90° condition; 1 for clear full-sentence explanation referencing vertically opposite.