Mathematics • Year 7 • Unit 3 • Lesson 8
Parallelograms and Rectangles
Build fluency with the parallelogram properties: opposite sides equal, opposite angles equal, co-interior angles add to 180°, diagonals bisect each other. A rectangle adds 4 right angles AND diagonals that are equal in length.
1. I do — fully worked example
Read every line. The working uses two parallelogram properties: opposite angles equal, and co-interior angles on the same side of a transversal sum to 180°.
Problem. In parallelogram ABCD, ∠A = 75°. Find ∠B, ∠C and ∠D.
Step 1 — Opposite angles equal.
∠C = ∠A = 75° (opp. angles of parallelogram)
Reason: in a parallelogram, the two angles at opposite corners are equal.
Step 2 — Adjacent angles are co-interior.
∠B = 180 − 75 = 105° (co-int. angles, AB ∥ DC)
Reason: ∠A and ∠B are on the same side of side AD, with AB parallel to DC; co-interior pairs sum to 180°.
Step 3 — Last angle uses opposite-equal again.
∠D = ∠B = 105° (opp. angles of parallelogram)
Check: 75 + 105 + 75 + 105 = 360° ✓ (∠ sum of quad).
Answer: ∠B = 105°, ∠C = 75°, ∠D = 105°.
2. We do — fill in the missing steps
Fill in each blank. The problem uses "diagonals of a parallelogram bisect each other" and "diagonals of a rectangle are equal". 4 marks
Problem. Rectangle WXYZ has diagonals meeting at M. WY = 26 cm. Find MX.
Step 1 — Diagonals of a rectangle are equal.
XZ = _______ cm (diagonals of rectangle equal)
Step 2 — Diagonals of a parallelogram bisect each other.
M is the midpoint of XZ, so MX = _______ × XZ.
Step 3 — Compute.
MX = _______ ÷ 2 = _______ cm.
Step 4 — Note.
All four half-diagonals WM, MY, XM, MZ are equal to _______ cm in a rectangle.
3. You do — independent practice
Use the right property for each problem and state the reason in brackets.
Foundation — one property at a time
3.1 In parallelogram ABCD, ∠A = 110°. Find ∠B. 1 mark
3.2 Parallelogram PQRS has PQ = 12 cm. Find SR. 1 mark
3.3 Rectangle WXYZ has diagonal WY = 20 cm. Find diagonal XZ. 1 mark
3.4 Diagonals of parallelogram ABCD meet at M with AM = 5 cm. Find MC. 1 mark
Standard — combine two properties
3.5 In parallelogram ABCD, ∠A = 65°. Find ∠B, ∠C and ∠D. State a reason for each. 2 marks
3.6 In parallelogram PQRS with diagonals meeting at M, PM = 9 cm and QM = 6 cm. Find diagonals PR and QS. 2 marks
Extension — push the property to the limit
3.7 Rectangle ABCD has AB = 24 cm and BC = 7 cm. The diagonals meet at M. Without using Pythagoras, explain what you CAN and CANNOT find about MA from the property "diagonals bisect each other" alone. 2 marks
3.8 Parallelogram ABCD has ∠A = (2x + 10)° and ∠B = (3x − 30)°. Use the co-interior rule to find x and state ∠A and ∠B. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (rectangle WXYZ, WY = 26)
Step 1: XZ = 26 cm (diagonals of rectangle equal).
Step 2: MX = ½ × XZ (diagonals bisect each other).
Step 3: MX = 26 ÷ 2 = 13 cm.
Step 4: All four halves = 13 cm in a rectangle.
3.1 — ∠A = 110°, find ∠B
Co-interior with ∠A: ∠B = 180 − 110 = 70° (co-int. angles, AB ∥ DC).
3.2 — PQ = 12 cm, find SR
Opposite sides equal: SR = 12 cm (opp. sides of parallelogram).
3.3 — WY = 20 cm, find XZ
Diagonals of rectangle equal: XZ = 20 cm.
3.4 — AM = 5 cm, find MC
Diagonals bisect each other: MC = 5 cm (diagonals of parallelogram bisect).
3.5 — Parallelogram, ∠A = 65°
∠C = 65° (opp. angles of parallelogram).
∠B = 180 − 65 = 115° (co-int. angles, AB ∥ DC).
∠D = 115° (opp. angles of parallelogram).
Check: 65 + 115 + 65 + 115 = 360° ✓
3.6 — PM = 9, QM = 6; find PR and QS
M is midpoint of PR (diagonals bisect), so MR = PM = 9 → PR = 9 + 9 = 18 cm.
M is midpoint of QS, so MS = QM = 6 → QS = 6 + 6 = 12 cm.
Note: PR ≠ QS — only EQUAL diagonals indicate a rectangle.
3.7 — Rectangle 24 × 7, what does "bisect" give?
From "diagonals bisect" alone, we know M is the midpoint of both diagonals, so MA = ½ × AC. We also know "diagonals of a rectangle are equal" so AC = BD. But the actual LENGTH of AC requires Pythagoras (24² + 7² = 625, AC = 25 cm). So bisection tells us MA equals half a diagonal, but NOT what that length is in cm. (Bonus answer: with Pythagoras, MA = 12.5 cm.)
3.8 — (2x + 10) + (3x − 30) = 180
Collect: 5x − 20 = 180 (co-int. angles, AB ∥ DC). 5x = 200, x = 40.
∠A = 2(40) + 10 = 90°, ∠B = 3(40) − 30 = 90°. Curiously, this parallelogram has 4 right angles — it's a rectangle.