Quadrilaterals · Ages 14–15
Conditions for a parallelogram
Opposite sides equal, opposite angles equal, diagonals bisect, one pair equal and parallel.
How to do it
Method
Opposite sides equal, opposite angles equal, diagonals bisect, one pair equal and parallel
AB // DC, 2x + 1 = x + 5, x = 4
Another way
Put it back in
2 × 4 + 1 = 9, 4 + 5 = 9
In full
A quadrilateral is a parallelogram if any one of these holds: both pairs of opposite sides parallel (the definition); both pairs of opposite sides equal (opp. sides equal); both pairs of opposite angles equal (opp. ∠s equal); diagonals bisecting each other (diags. bisect each other); one pair of opposite sides equal and parallel (opp. sides equal and //). Not enough: one pair parallel and the other pair equal can be an isosceles trapezium; one pair of sides and one pair of angles equal needn't be either. The method: see what you're given and check it against each condition; a condition that says both pairs needs both. Finding an unknown: AB = (2x + 1) cm, DC = (x + 5) cm and AB // DC; for a parallelogram AB = DC: 2x + 1 = x + 5, x = 4. Another way: put it back in, x = 4 gives AB = DC = 9 cm.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
In ABCD, AB // DC, AB = (2x + 5) cm and DC = (4x − 5) cm. For ABCD to be a parallelogram (opp. sides equal and //), x =
Show the answer and working
Answer: 5
AB is already parallel to DC, so it needs AB = DC: 2x + 5 = 4x − 5, x = 5.
Example 2
By the marks in the picture, ABCD is a parallelogram. Why?
- A. opp. ∠s equal
- B. opp. sides equal
- C. diags. bisect each other
- D. opp. sides equal and //
Choose the right answer
Show the answer and working
Answer: D. opp. sides equal and //
AB and DC carry the same parallel marks and the same equal marks: one pair of opposite sides equal and parallel.
Example 3
The diagonals of ABCD cross at E, with BE = ED = 6 cm, AE = (x + 4) cm and EC = (2x − 4) cm. For ABCD to be a parallelogram, x =
Show the answer and working
Answer: 8
BD is already halved; it needs AE = EC too (diags. bisect each other): x + 4 = 2x − 4, x = 8. Check: AE = EC = 12 cm.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.By the marks in the picture, ABCD is a parallelogram. Why?
- A. diags. bisect each other
- B. opp. sides equal
- C. opp. sides equal and //
- D. opp. ∠s equal
Choose the right answer
Answer:
2.In ABCD, AB // DC, AB = (2x + 7) cm and DC = (5x − 20) cm. For ABCD to be a parallelogram (opp. sides equal and //), x =
Answer:
3.By the marks in the picture, ABCD is a parallelogram. Why?
- A. opp. ∠s equal
- B. opp. sides equal and //
- C. opp. sides equal
- D. diags. bisect each other
Choose the right answer
Answer:
4.The diagonals of ABCD cross at E, with BE = ED = 5 cm, AE = (x + 6) cm and EC = (2x − 6) cm. For ABCD to be a parallelogram, x =
Answer:
5.In the picture, AB // DC and AD = BC. Must ABCD be a parallelogram?
- A. not necessarily: it can be an isosceles trapezium
- B. yes, because opposite sides are equal
- C. yes, because opposite angles are equal
- D. yes, because one pair is equal and parallel
Choose the right answer
Answer:
All the answers
- 1. B. opp. sides equal
- 2. 9
- 3. D. diags. bisect each other
- 4. 12
- 5. A. not necessarily: it can be an isosceles trapezium
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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