To test if a given quadrilateral is a parallelogram, do we have to check that the opposite sides are parallel? Are there other ways to test this?
Step-by-step solution
Idea: The properties of a parallelogram (opposite sides equal, opposite angles equal, diagonals bisecting each other) all have true converses, so each of them can be used as a test. One more test uses a single pair of sides that is both equal and parallel.
- By definition, a parallelogram has both pairs of opposite sides parallel. Checking this directly is one test, but not the only one.½ mark
- Other tests (each is enough on its own): (1) both pairs of opposite sides are equal; (2) both pairs of opposite angles are equal; (3) the diagonals bisect each other; (4) one pair of opposite sides is equal and parallel.1 mark
- Lengths and angles are often easier to measure than parallelism, so these tests are very useful in proofs and in practice.½ mark
Answer to write in the exam
Not necessary to check that both pairs of opposite sides are parallel
ABCD is a parallelogram if: (1) AB = CD and AD = BC, or (2) ∠A = ∠C and ∠B = ∠D,
or (3) diagonals AC and BD bisect each other, or (4) AB = CD and AB ‖ CD
∴ Yes, there are other tests.
Common mistakes that cost marks
- Using only one pair of equal opposite sides (without parallel). An isosceles trapezium has one pair of equal sides but is not a parallelogram.
- Using “one pair of opposite angles equal”. Both pairs are needed.
- Using “diagonals are equal”. That does not make a parallelogram (an isosceles trapezium has equal diagonals).
How this can come in the exam
Which condition is NOT enough to show that quadrilateral ABCD is a parallelogram?
- AB = CD and AD = BC
- AB ‖ CD and AB = CD
- Diagonals AC and BD bisect each other
- AB = CD and AD ‖ BC
Show answer
(D) AB = CD and AD ‖ BC
AB = CD with the other pair AD ‖ BC describes an isosceles trapezium too. The other three are standard tests.
Try one yourself
In quadrilateral PQRS, ∠P = ∠R = 110° and ∠Q = ∠S = 70°. Is PQRS a parallelogram?
Show answer
Yes: both pairs of opposite angles are equal, which is enough. (Also adjacent angles add to 180°.)
More questions like this
- Recall the following properties of a parallelogram that we proved last year.
- 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?
- If the opposite sides of a quadrilateral are of equal length, then it is a parallelogram.
- 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.
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