Mathematics • Year 8 • Unit 3 • Lesson 13
Angles — Mixed Challenge
Pull everything from Lesson 13 together: classifying angles, complements, supplements, vertically opposite pairs, angles at a point and algebraic ratios. Six mixed problems, one "find the mistake", and one open-ended angle-at-a-point design challenge.
1. Mixed problems — choose the right move
Each question pulls a different idea from Lesson 13. Always state the rule (complementary / supplementary / vert. opp. / angles at a point) before you calculate. 3 marks each
1.1 Classify each of these angles (acute / right / obtuse / straight / reflex): (i) 95°, (ii) 90°, (iii) 270°, (iv) 75°.
1.2 Find the complement of 18° and the supplement of 18°.
1.3 Two straight lines cross. One angle is 72°. Find the other three angles, naming the rule for each.
1.4 Four angles meet at a point: 110°, 85°, 90° and x. Find x.
1.5 Two complementary angles are in the ratio 2 : 7. Find each angle.
1.6 Two angles on a straight line are (3x + 10)° and (2x − 5)°. Set up an equation and find x. Then find both angles.
2. Find the mistake
A Year 8 student has tried to find the missing angle at a point. Three angles around a point are 85°, 95°, and x. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — find x at a point with 85°, 95° and x:
Line 1: Angles at a point sum to 180°.
Line 2: 85 + 95 + x = 180
Line 3: 180 + x = 180
Line 4: So x = 0°.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Revisit lesson § Card 8 — angles at a point sum to 360°, not 180°. (180° is angles on a straight line.)3. Open-ended challenge — design a 360° fan
This question has more than one valid answer. 4 marks
3.1 A circular fan has blades that meet at the central hub. The angles between consecutive blades must sum to 360° (angles at a point).
Design three different fans with the following constraints:
Fan A: 4 blades, all equal-spaced.
Fan B: 5 blades, all equal-spaced.
Fan C: 3 blades, NOT equal-spaced. Choose any three angles between consecutive blades that sum to 360°, and one of the angles must be obtuse.
For each fan:
(i) State each angle between consecutive blades.
(ii) Show the sum check.
(iii) Classify each angle (acute / right / obtuse / reflex).
Bonus: Fan C must have at least one acute AND at least one obtuse angle (so the blades are clearly uneven).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Classify the angles
(i) 95° = obtuse (between 90° and 180°).
(ii) 90° = right.
(iii) 270° = reflex (between 180° and 360°).
(iv) 75° = acute (less than 90°).
1.2 — Complement and supplement of 18°
Complement = 90 − 18 = 72°. Supplement = 180 − 18 = 162°.
1.3 — Lines crossing, one angle 72°
Vertically opposite = 72° (vert. opp. ∠s equal).
Adjacent angles = 180 − 72 = 108° each (angles on a straight line).
Four angles: 72°, 108°, 72°, 108°.
1.4 — Angles at a point sum to 360°
110 + 85 + 90 + x = 360. 285 + x = 360. x = 75°.
1.5 — Complementary ratio 2 : 7
Total parts = 2 + 7 = 9. 1 part = 90 ÷ 9 = 10°. Angles: 2 × 10 = 20° and 7 × 10 = 70°. Check: 20 + 70 = 90 ✓.
1.6 — Algebra on a straight line
(3x + 10) + (2x − 5) = 180 → 5x + 5 = 180 → 5x = 175 → x = 35.
Angle 1 = 3(35) + 10 = 115°. Angle 2 = 2(35) − 5 = 65°. Check: 115 + 65 = 180 ✓.
2 — Find the mistake
(a) The mistake is on Line 1 (and carries into Lines 2, 3, 4).
(b) The student used the wrong total. Angles at a point sum to 360°, not 180°. (180° is for angles on a straight line, which only fills half a turn.)
(c) Corrected working:
Angles at a point sum to 360°.
85 + 95 + x = 360
180 + x = 360
x = 175°. ✓
Sanity check: x ≈ 175° is just under a straight line — visually, this is the "big leftover" gap that closes the full turn around the point.
3 — Fan design (sample solutions)
Fan A — 4 equal blades: 90° + 90° + 90° + 90° = 360° ✓. All four angles are right angles.
Fan B — 5 equal blades: 360 ÷ 5 = 72° each. Sum: 5 × 72 = 360° ✓. All five are acute.
Fan C — 3 unequal blades (sample): 80° + 130° + 150° = 360° ✓. Angles: 80° (acute), 130° (obtuse), 150° (obtuse). Has at least one acute and at least one obtuse.
Other valid Fan C answers: 60° + 120° + 180° (but 180° is straight, edge case); 50° + 100° + 210° ✗ (210° is reflex, not a "blade angle"); 70° + 110° + 180° — straight; 90° + 120° + 150° ✓ (right + 2 obtuse).
Marking: 1 mark for Fan A correct (90° × 4); 1 mark for Fan B correct (72° × 5); 1 mark for Fan C with three valid angles summing to 360°; 1 bonus mark if Fan C has both an acute AND an obtuse angle (and avoids reflex angles, which wouldn't physically fit between blades).