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Tests for a parallelogram · 3 marks

If the converse of any of the three properties is true, it can be used as an alternate way to show that a quadrilateral is a parallelogram. To explore this, let us write the converses. Can you experiment and guess what the answers are?

  1. (a) Converse: If the opposite sides of a quadrilateral are equal, then it is a parallelogram. Is this converse statement true?
  2. (b) Converse: If the opposite angles of a quadrilateral are equal, then it is a parallelogram. Is this converse statement true?
  3. (c) Converse: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Is this converse statement true?
Answer: All three converses are true: a quadrilateral is a parallelogram if (a) its opposite sides are equal, or (b) its opposite angles are equal, or (c) its diagonals bisect each other.

Step-by-step solution

Idea: A converse swaps the “if” and “then” parts. Experiment with sticks or drawings, then prove each one by running the proof of the original property backwards.

(a) Converse: If the opposite sides of a quadrilateral are equal, then it is a parallelogram. Is this converse statement true?

  1. True. If AB = CD and AD = BC, then with diagonal AC common, ∆ACD ≅ ∆CAB (SSS). So ∠CAB = ∠ACD, which are alternate angles for AB and DC with transversal AC: AB ‖ DC. Likewise ∠ACB = ∠CAD gives AD ‖ BC.1 mark
True.

(b) Converse: If the opposite angles of a quadrilateral are equal, then it is a parallelogram. Is this converse statement true?

  1. True. If ∠A = ∠C and ∠B = ∠D, then 360° = 2(∠A + ∠B), so ∠A + ∠B = 180°: co-interior angles on transversal AB, so AD ‖ BC. Similarly ∠B + ∠C = 180° gives AB ‖ DC.1 mark
True.

(c) Converse: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Is this converse statement true?

  1. True. If the diagonals meet at E with EA = EC and EB = ED, then ∆AED ≅ ∆CEB (SAS, vertically opposite angles at E). So ∠DAE = ∠BCE, alternate angles: AD ‖ BC. Likewise ∆EAB ≅ ∆ECD gives AB ‖ DC.1 mark
True.
All three converse statements are true, so each gives a test for a parallelogram.

Answer to write in the exam

(a)

AB = CD, AD = BC, AC common ⇒ ∆ACD ≅ ∆CAB (SSS)

⇒ ∠CAB = ∠ACD and ∠ACB = ∠CAD (CPCT) ⇒ AB ‖ DC, AD ‖ BC (alternate angles equal)

∴ True: ABCD is a parallelogram.

(b)

∠A + ∠B + ∠C + ∠D = 360° and ∠A = ∠C, ∠B = ∠D ⇒ ∠A + ∠B = 180° = ∠B + ∠C

⇒ AD ‖ BC and AB ‖ DC (co-interior angles supplementary)

∴ True: ABCD is a parallelogram.

(c)

EA = EC, EB = ED, ∠AED = ∠CEB (vertically opposite) ⇒ ∆AED ≅ ∆CEB (SAS)

⇒ ∠DAE = ∠BCE ⇒ AD ‖ BC; similarly ∆EAB ≅ ∆ECD ⇒ AB ‖ DC

∴ True: ABCD is a parallelogram.

Common mistakes that cost marks

  • Thinking a converse is automatically true because the original is true. Each converse needs its own proof (for example, “a square has equal diagonals” has a false converse).
  • In (b), using only one pair of equal opposite angles. Both pairs are needed.
  • In (a), using ASA as in the original proof. Here only sides are known, so the test is SSS.

How this can come in the exam

Assertion–Reason (1 mark)

Assertion (A): A quadrilateral with sides 5 cm, 8 cm, 5 cm, 8 cm taken in order is a parallelogram.
Reason (R): If both pairs of opposite sides of a quadrilateral are equal, it is a parallelogram.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
Taken in order, the opposite sides are 5 and 5, 8 and 8: both pairs equal, so by R it is a parallelogram.

Try one yourself

The diagonals of quadrilateral KLMN meet at O with OK = 4, OM = 4, OL = 7, ON = 7. What kind of quadrilateral is KLMN?

Show answer

The diagonals bisect each other (O is the midpoint of KM and of LN), so KLMN is a parallelogram.

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