Mathematics • Year 7 • Unit 3 • Lesson 10

Angle Sum of Quadrilaterals — Real World

Use the 360° angle sum to find a missing corner angle in field plans, blueprints, kite designs, paddock surveys and origami. State the reason (∠ sum of quad) after each line of working.

Apply · Real-World Maths

1. Word problems

Read carefully. Identify the four corner angles of the shape, then use the 360° rule with a reason in brackets.

1.1 — Paddock survey. A surveyor measures three corner angles of a four-sided paddock as 85°, 95° and 110°.

(a) Find the fourth corner angle.
(b) State the reason in brackets. 2 marks

Stuck? 360° − sum of the three known angles.

1.2 — Origami crane base. The flat base of an origami crane is a quadrilateral whose angles are in the ratio 1 : 2 : 3 : 4. Let the angles be x°, 2x°, 3x°, 4x°.

(a) Write the equation and state the reason.
(b) Solve for x.
(c) State the four angles. 3 marks

Stuck? Sum of ratios is 1 + 2 + 3 + 4 = 10, so 10x = 360.

1.3 — Backyard pool plan. A backyard pool is a quadrilateral with angles (3x − 10)°, (2x)°, (x + 30)° and (2x + 10)°.

(a) Set up the angle-sum equation.
(b) Solve for x.
(c) State each pool corner angle. 3 marks

Stuck? Collect x coefficients (3 + 2 + 1 + 2 = 8) and constants (−10 + 30 + 10 = 30) separately.

1.4 — Festival kite. A festival kite has angles 80°, 130°, 80° and x°.

(a) Find x.
(b) Identify which special quadrilateral fits this pattern of angles (one pair of equal opposite angles + 360° sum). 2 marks

Stuck? Two equal angles in a kite are the "between unequal sides" pair.

1.5 — Architect's quirky window. An architect designs a four-sided window with three angles labelled 100°, 80° and 120°. The fourth angle is left blank for the joiner to calculate.

(a) Find the missing fourth angle.
(b) Could the window be a parallelogram? Justify your answer using opposite-angle pairs. 3 marks

Stuck? In a parallelogram, opposite angles are EQUAL. Look at the four angles you've found — are they in two equal pairs?

2. Explain your thinking

Use full sentences. 4 marks

2.1 A classmate looks at a quadrilateral and writes "the angles must sum to 180° because that's the angle rule for shapes." Explain (i) what the correct angle sum for a quadrilateral is, (ii) WHY it's that value (use the "split with a diagonal" proof), and (iii) how to spot when someone has confused the triangle rule with the quadrilateral rule.

Stuck? Triangle = 180°, quadrilateral = 360°. The proof: one diagonal makes 2 triangles, and 2 × 180 = 360.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Paddock 85°, 95°, 110°, ?

(a) Fourth = 360 − 85 − 95 − 110 = 70° (∠ sum of quad).
(b) Reason: (∠ sum of quad) — the four interior angles of any quadrilateral sum to 360°.

1.2 — Origami 1 : 2 : 3 : 4

(a) x + 2x + 3x + 4x = 360 (∠ sum of quad).
(b) 10x = 360, so x = 36.
(c) Angles = 36°, 72°, 108°, 144°. Check: 36 + 72 + 108 + 144 = 360° ✓.

1.3 — Pool (3x − 10) + 2x + (x + 30) + (2x + 10) = 360

(a) Equation: 8x + 30 = 360 (∠ sum of quad).
(b) 8x = 330, so x = 330 ÷ 8 = 41.25. (Accept x = 165/4.)
(c) Angles = 3(41.25) − 10 = 113.75°; 2(41.25) = 82.5°; 41.25 + 30 = 71.25°; 2(41.25) + 10 = 92.5°. So angles ≈ 113.75°, 82.5°, 71.25°, 92.5°. Check: 113.75 + 82.5 + 71.25 + 92.5 = 360° ✓.

1.4 — Festival kite, 80°, 130°, 80°, x°

(a) x = 360 − 80 − 130 − 80 = 70° (∠ sum of quad). Angles: 80°, 130°, 80°, 70°.
(b) Two equal opposite angles (the pair of 80°s) is characteristic of a kite. So this fits a kite. (NOTE: in a real kite layout the equal pair should be opposite each other — labelled at vertices B and D — and the other two angles are different, exactly as here.)

1.5 — Window 100°, 80°, 120°, ?

(a) Fourth = 360 − 100 − 80 − 120 = 60° (∠ sum of quad).
(b) For a parallelogram, opposite angles must be equal — the angles need to fall into two equal pairs. Here we have 100°, 80°, 120°, 60° — no two of those are equal, so the four angles do NOT pair up. The window cannot be a parallelogram.

2.1 — Explain your thinking (sample response)

(i) The correct angle sum for a quadrilateral is 360°, not 180°. 180° is the angle sum for a triangle.
(ii) Proof: draw one diagonal across the quadrilateral. The diagonal splits the shape into TWO triangles. Each triangle has angle sum 180°, and the angles of the two triangles together make up exactly the four interior angles of the quadrilateral. So total = 2 × 180° = 360°.
(iii) Tell signs the rule is being mixed up: (a) the working sets up an equation like "a + b + c + d = 180" — that's wrong because there are four angles in a quadrilateral; (b) a missing-fourth-angle calculation comes out negative or implausibly small (e.g. 180 − 95 − 95 − 85 = −95°) — that's a giveaway that 180° was used instead of 360°.

Marking: 1 for stating 360°; 1 for the diagonal/two-triangle proof; 1 for at least one diagnostic warning sign; 1 for clear writing.