If the opposite sides of a quadrilateral are of equal length, then it is a parallelogram.
Step-by-step solution
To find: Prove that ABCD is a parallelogram (AB ‖ DC and AD ‖ BC)
Idea: Run the proof that “opposite sides of a parallelogram are equal” backwards: the same two triangles ∆ACD and ∆CAB, but now proved congruent by SSS, and the equal angles then give the parallel sides.
- Join the diagonal AC. In ∆ACD and ∆CAB: CD = AB (given), DA = BC (given), AC = CA (common).1 mark
- So ∆ACD ≅ ∆CAB (SSS). Corresponding angles are equal: ∠ACD = ∠CAB and ∠CAD = ∠ACB.1 mark
- ∠CAB and ∠ACD are alternate angles made by transversal AC with AB and DC; they are equal, so AB ‖ DC. Similarly ∠CAD = ∠ACB gives AD ‖ BC. Both pairs of opposite sides are parallel: ABCD is a parallelogram.1 mark
Answer to write in the exam
Join AC. In ∆ACD and ∆CAB:
CD = AB, DA = BC (given), AC = CA (common)
∴ ∆ACD ≅ ∆CAB (SSS)
⇒ ∠ACD = ∠CAB and ∠CAD = ∠ACB (CPCT)
⇒ AB ‖ DC and AD ‖ BC (alternate angles equal)
∴ ABCD is a parallelogram.
Common mistakes that cost marks
- Matching the triangles in the wrong order (∆ACD ≅ ∆ABC). Write ∆ACD ≅ ∆CAB so that D matches B.
- Concluding “AB ‖ DC” from AB = DC directly. Equal lengths alone do not give parallel lines; the equal alternate angles do.
- Using only one pair of equal opposite sides. An isosceles trapezium has one pair equal but is not a parallelogram.
How this can come in the exam
Two sticks of 12 cm and two sticks of 7 cm are joined end to end to make a four-sided frame, with the two 12 cm sticks opposite each other. Show that the frame is always a parallelogram, however it is pushed.
Show answer
Opposite sides are 12 cm and 12 cm, 7 cm and 7 cm, so both pairs of opposite sides are equal (1 mark). A quadrilateral with both pairs of opposite sides equal is a parallelogram (SSS on a diagonal gives equal alternate angles), so every shape of the frame is a parallelogram (1 mark).Try one yourself
In quadrilateral PQRS, PQ = 6 cm, QR = 4 cm, RS = 6 cm and SP = 4 cm. Prove that PQ ‖ RS.
Show answer
Join PR. ∆PQR ≅ ∆RSP (SSS: PQ = RS, QR = SP, PR common), so ∠QPR = ∠SRP (CPCT). These are alternate angles for PQ and RS with transversal PR, so PQ ‖ RS.
More questions like this
- If the opposite angles of a quadrilateral are equal, then it is a parallelogram.
- The angles of a quadrilateral add up to 360°. Therefore, if the opposite angles are equal, what can we say about adjacent angles? Is the converse of your answer true? Conclude that the result “if the opposite angles of a quadrilateral are equal, then it is a parallelogram” can also be stated as follows. “If each pair of adjacent angles in a quadrilateral ABCD …then ABCD is a parallelogram.” Fill in the blank.
- A quadrilateral whose diagonals bisect each other is a parallelogram.
- A quadrilateral with one pair of equal and parallel opposite sides is a parallelogram.
- Suppose in a quadrilateral ABCD we have AB ‖ DC and AB = DC. Let E be the intersection point of the diagonals AC and BD. (Why must the diagonals intersect?)
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