Four right angles

The idea

If all the angles of a quadrilateral are right angles, its opposite sides are equal, so a rectangle can be defined simply as a quadrilateral whose angles are all 90°.

One test is enough?

To be a rectangle, a quadrilateral needs right angles and equal opposite sides. But maybe the right angles alone are enough?

A quadrilateral has four right angles. Could its opposite sides be unequal?

• Yes, they could be
• No, they must be equal
• I cannot tell

Walk round and come back

A walker starts at A, walks a side, turns left by 90°, walks the next side, and so on: four sides, four right-angle turns. Choose the lengths. Can you make the walker end back at A?

The walk only closes one way

If all the angles of a quadrilateral are right angles, its opposite sides are equal, so a rectangle can be defined simply as a quadrilateral whose angles are all 90°.

Why? Join BD. ∠ABD = ∠CDB (each is 90° − ∠DBC), the angles at A and C are 90°, and BD is common. So ∆BAD ≅ ∆DCB by AAS: AD = CB and DC = BA.

Notes

If all the angles of a quadrilateral are right angles, its opposite sides are equal. So a rectangle is simply a quadrilateral whose angles are all 90°.

Check yourself

In quadrilateral PQRS all four angles are 90°. PQ = 9 cm and QR = 4 cm. How long is SR?

ABCD has four right angles. AB = 7 cm and BC = 3 cm. What is its perimeter (the sum of all four sides)?

Answer: 20 cm

CD = AB = 7 cm and DA = BC = 3 cm. The perimeter is 7 + 3 + 7 + 3 = 20 cm.

A quadrilateral has four angles of 90°. AB = 6 cm, BC = 2.5 cm, CD = x cm, DA = y cm. Find x + y.

Answer: 8.5 cm

CD is opposite AB, so x = 6. DA is opposite BC, so y = 2.5. x + y = 8.5.

Why can the definition of a rectangle simply say “all four angles are 90°”?

• 4 cm. SR is opposite PQ, not QR. In a quadrilateral with four right angles, opposite sides are equal.
• 9 cm — correct. Yes! SR is opposite PQ, so SR = PQ = 9 cm.
• 13 cm. 13 is 9 + 4. SR is opposite PQ, so it equals PQ.
• It can be any length. With four right angles the walk must close, and that fixes SR = PQ.
• Because the sides of a rectangle do not matter. The sides do matter. They just do not need a separate condition.
• Because equal opposite sides then follow automatically — correct. Yes! With four right angles the walk can only close if opposite sides are equal.
• Because every rectangle is a square. Squares are special rectangles. Most rectangles are not squares.