Mathematics • Year 8 • Unit 3 • Lesson 13
Angles in the Real World
Use complementary, supplementary, vertically opposite and angle-at-a-point rules where they actually appear: clock hands, road junctions, ramps, picture frames and pizza slices. Then explain your thinking.
1. Word problems
Each problem hides an angle relationship. Sketch and label, name the rule, then calculate. Show working — a single answer with no working earns only half marks.
1.1 — Clock hand angle. At 3:00, the angle between the minute and hour hands is exactly 90°. At 2:00, the angle is 60° (clockwise from 12).
(a) Find the complement of 60°.
(b) Find the supplement of 60°.
(c) At 5:00 the angle between the hands is 150°. Classify it (acute / right / obtuse / reflex) and find its supplement. 3 marks
1.2 — Road junction. Two straight roads cross at a city intersection. One of the four angles formed is 65°.
(a) Find the vertically opposite angle. State the rule.
(b) Find the two other (adjacent) angles. State the rule.
(c) Do all four angles add to 360°? Show the check. 3 marks
1.3 — Wheelchair ramp. Australian Standards say a wheelchair ramp should make an angle of no more than 8° with the horizontal (the "ramp angle").
(a) The "rise angle" plus the "ramp angle" together form a right angle (with the vertical post). If the ramp angle is 8°, what is the rise angle?
(b) Classify both angles (acute/right/obtuse).
(c) Is a 12° ramp acceptable? Justify in one sentence. 3 marks
1.4 — Picture frame. A rectangular picture frame is built from four mitred corners. At each corner, the two diagonal cuts must meet to form a 90° angle.
(a) If each cut is at 45°, show that the two cuts together form a right angle.
(b) A worker cuts one piece at 47° by mistake. What angle should the matching piece be cut at so the corner is still 90°?
(c) What relationship are the two cuts in? (Complementary / supplementary / vertically opp.) 3 marks
1.5 — Pizza slices. A pizza is cut into 6 equal slices from the centre. Then one cut is shifted so that one slice is twice the size of the others.
(a) Originally, what is the angle of each of the 6 equal slices? (Hint: 360° divided equally.)
(b) If one slice is twice the size, and the other 5 are equal, find each angle. (Let small slice = x, large = 2x. Set up an equation.)
(c) Confirm all 6 angles sum to 360°. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate is asked to find the complement of 110°. They write "complement = 90 − 110 = −20°." In your own words, explain (i) what mistake they have made, (ii) why a complement of 110° is actually impossible, and (iii) what the supplement of 110° would be instead, and what the relationship looks like in a diagram. Use the phrase "C for Corner, S for Straight" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Clock hands
(a) Complement of 60° = 90 − 60 = 30°.
(b) Supplement of 60° = 180 − 60 = 120°.
(c) 150° is obtuse (between 90° and 180°). Supplement = 180 − 150 = 30°.
1.2 — Road junction
(a) Vertically opposite = 65° (vert. opp. angles equal).
(b) Adjacent angles = 180 − 65 = 115° each (angles on a straight line).
(c) Sum check: 65 + 115 + 65 + 115 = 360° ✓ — confirms angles at a point.
1.3 — Wheelchair ramp
(a) Rise angle = 90 − 8 = 82° (complementary).
(b) Both are acute (less than 90°).
(c) Not acceptable — 12° > 8°, so the ramp is too steep under Australian Standards.
1.4 — Picture frame
(a) 45 + 45 = 90 ✓ — the two 45° cuts form a right-angled corner.
(b) Matching cut = 90 − 47 = 43°.
(c) Complementary — the two cuts sum to 90° to form a right angle.
1.5 — Pizza slices
(a) 6 equal slices: 360 ÷ 6 = 60° each.
(b) Let small slice = x, large = 2x. 5x + 2x = 7x = 360°, so x = 360 ÷ 7 ≈ 51.4° (5 small slices). Large slice = 2x ≈ 102.9°.
(c) Check: 5 × 51.4 + 102.9 = 257.0 + 102.9 = 359.9° ≈ 360° ✓ (rounding).
2.1 — Explain your thinking (sample response)
The classmate has made the mistake of subtracting from 90° even though 110° is bigger than 90°. A complement only exists when the angle is less than 90°, because C for Corner, S for Straight — the complement is what you add to make a right angle (90°), and you can't add a negative angle. Since 110° is already obtuse (greater than 90°), it has no complement. What they probably wanted is the supplement of 110°, which is 180 − 110 = 70°. In a diagram, two adjacent supplementary angles form a straight line: a 110° angle on one side and a 70° angle on the other, lying along a single straight ray.
Marking: 1 mark for spotting the "applied 90° rule to an obtuse angle" mistake; 1 mark for explaining "complement only exists when angle < 90°"; 1 mark for stating the supplement = 70° and describing the straight-line diagram; 1 mark for using the phrase "C for Corner, S for Straight" in a clear sentence.