Mathematics • Year 7 • Unit 3 • Lesson 8
Parallelograms and Rectangles — Mixed Challenge
Mix every property of parallelograms and rectangles: opposite sides equal, opposite angles equal, co-interior 180°, diagonals bisect each other, diagonals of rectangle equal. Spot a property mix-up, then create a "minimum-information" puzzle.
1. Mixed problems
Decide which property to use, then show working with a reason in brackets. 2 marks each
1.1 Parallelogram ABCD has ∠A = 120°. Find ∠B, ∠C and ∠D.
1.2 Parallelogram PQRS has PQ = 15 cm and PS = 9 cm. Find QR and SR.
1.3 Rectangle WXYZ has diagonal WY = 17 cm. Find diagonal XZ and the distance from the centre M to W.
1.4 In parallelogram ABCD, ∠A is twice ∠B. Find both ∠A and ∠B.
1.5 The diagonals of parallelogram EFGH meet at M with EM = 8 cm and FM = 5 cm. Find EG and FH.
1.6 A quadrilateral has diagonals that BISECT each other but are NOT equal in length. Choose the most specific name for the shape from {parallelogram, rectangle, rhombus, square} and justify in one sentence.
2. Find the mistake
A Year 7 student worked out the angles of parallelogram ABCD given ∠A = 80°. Their working is shown. Exactly one line is wrong. Spot it, explain why, then redo the working. 3 marks
Student's working — parallelogram ABCD with ∠A = 80°:
Line 1: All angles in a parallelogram are equal, so ∠B = ∠C = ∠D = 80°.
Line 2: Check: 80 + 80 + 80 + 80 = 320°.
Line 3: Hmm, that's not 360°. Final answer: ∠B = ∠C = ∠D = 80° (the rule must be wrong).
(a) Which line contains the conceptual error?
(b) Explain in one or two sentences why "all angles equal" is wrong for a general parallelogram.
(c) Write out the corrected working.
Stuck? Only OPPOSITE angles are equal; adjacent angles are co-interior (180°). A parallelogram with 4 equal angles would have to be a rectangle.3. Open-ended challenge — minimum-information puzzle
This question has more than one correct answer. 4 marks
3.1 Design a parallelogram (or rectangle) puzzle where the SINGLE given measurement is enough to find every other angle AND every other length labelled on the diagram.
Requirements: (i) draw a clearly labelled parallelogram or rectangle ABCD with diagonals meeting at M; (ii) give exactly ONE measurement (either an angle or a length); (iii) work out every other labelled angle and length, with a property in brackets after each line; (iv) explain why one number is enough.
Hint: An angle measurement unlocks all four angles via opposite-equal + co-interior. A side measurement unlocks the opposite side. A half-diagonal unlocks the full diagonal via bisection (and, in a rectangle, the OTHER diagonal too).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Parallelogram ABCD, ∠A = 120°
∠C = 120° (opp. angles of parallelogram).
∠B = 180 − 120 = 60° (co-int. angles, AB ∥ DC).
∠D = 60° (opp. angles). Check: 120 + 60 + 120 + 60 = 360° ✓.
1.2 — PQ = 15, PS = 9
QR = PS = 9 cm, SR = PQ = 15 cm (opp. sides of parallelogram).
1.3 — Rectangle WXYZ, WY = 17
XZ = 17 cm (diagonals of rectangle equal). MW = ½ × WY = 8.5 cm (diagonals bisect each other).
1.4 — ∠A = 2 × ∠B
Adjacent angles are co-interior: ∠A + ∠B = 180. Substituting ∠A = 2∠B: 2∠B + ∠B = 180, so 3∠B = 180 and ∠B = 60°. Then ∠A = 120° (co-int. angles, AB ∥ DC).
1.5 — EM = 8, FM = 5
M is the midpoint of EG (diagonals bisect): MG = EM = 8 → EG = 16 cm.
M is the midpoint of FH: MH = FM = 5 → FH = 10 cm.
1.6 — Bisecting but UNEQUAL diagonals
Most specific name: parallelogram. Reason: bisecting diagonals identify a parallelogram, but equal-length diagonals would force it to be a rectangle (and the question rules that out).
2 — Find the mistake
(a) The error is on Line 1.
(b) In a parallelogram, only OPPOSITE angles are equal — not all four. Adjacent angles are co-interior, so they add to 180°, not equal. If all four angles WERE equal, each would be 90° and the parallelogram would be a rectangle.
(c) Correct working (∠A = 80°):
∠C = ∠A = 80° (opp. angles of parallelogram).
∠B = 180 − 80 = 100° (co-int. angles, AB ∥ DC).
∠D = ∠B = 100° (opp. angles).
Check: 80 + 100 + 80 + 100 = 360° ✓.
3 — Open-ended challenge (sample solution)
Sample design. Rectangle ABCD with diagonals AC and BD meeting at M. Single given measurement: AM = 6 cm.
BM = AM = 6 cm (diagonals of rectangle equal AND bisect each other, so all four half-diagonals are equal). Similarly CM = DM = 6 cm.
AC = AM + MC = 6 + 6 = 12 cm (diagonals bisect).
BD = AC = 12 cm (diagonals of rectangle equal).
All four angles of ABCD = 90° (rectangle by definition).
Why one number is enough: in a rectangle, the diagonals are equal AND bisect each other, so every half-diagonal equals AM. The angles are pinned down by the rectangle definition (4 right angles).
Marking: 1 for a clear labelled diagram with ONE given; 1 for using the correct properties; 1 for finding all the labelled quantities; 1 for the "why one number is enough" justification. Accept any valid design where one number unlocks every labelled quantity.