Geometry and proof · Ages 14–15

Conditions for a parallelogram worksheet

Part of: Quadrilaterals

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Geometry and proof · Ages 14–15

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.In ABCD, AB // DC, AB = (3x + 7) cm and DC = (6x − 20) cm. For ABCD to be a parallelogram (opp. sides equal and //), x =
Answer:
2.In ABCD, AB // DC, AB = (2x + 3) cm and DC = (4x − 7) cm. For ABCD to be a parallelogram (opp. sides equal and //), x =
Answer:
3.In ABCD, AB // DC, AB = (3x + 7) cm and DC = (5x − 11) cm. For ABCD to be a parallelogram (opp. sides equal and //), x =
Answer:
4.The diagonals of ABCD cross at E, with BE = ED = 3 cm, AE = (x + 1) cm and EC = (2x − 6) cm. For ABCD to be a parallelogram, x =
Answer:
5.The diagonals of ABCD cross at E, with BE = ED = 5 cm, AE = (x + 4) cm and EC = (2x − 1) cm. For ABCD to be a parallelogram, x =
Answer:

6.By the marks in the picture, ABCD is a parallelogram. Why?
- A. opp. sides equal and //
- B. diags. bisect each other
- C. opp. ∠s equal
- D. opp. sides equal
Choose the right answer
Answer:
7.In the quadrilateral ABCD, ∠A = 100°, ∠B = 80° and ∠C = 100°. ABCD must be a parallelogram.
True or false?
Answer:
8.In the quadrilateral ABCD, ∠A = 105°, ∠B = 75° and ∠C = 120°. ABCD must be a parallelogram.
True or false?
Answer:
9.In the quadrilateral ABCD, ∠A = 85°, ∠B = 95° and ∠C = 90°. ABCD must be a parallelogram.
True or false?
Answer:
10.By the marks in the picture, ABCD is a parallelogram. Why?
- A. opp. sides equal
- B. opp. ∠s equal
- C. diags. bisect each other
- D. opp. sides equal and //
Choose the right answer
Answer:
11.By the marks in the picture, ABCD is a parallelogram. Why?
- A. opp. sides equal
- B. diags. bisect each other
- C. opp. ∠s equal
- D. opp. sides equal and //
Choose the right answer
Answer:
12.In the picture, AB // DC and AD = BC. Must ABCD be a parallelogram?
- A. yes, because one pair is equal and parallel
- B. yes, because opposite sides are equal
- C. yes, because opposite angles are equal
- D. not necessarily: it can be an isosceles trapezium
Choose the right answer
Answer:

13.A carpenter measures a frame ABCD: AB = CD = 61 cm and AD = BC = 41 cm. He says it must be 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
Working:
Answer:
14.A worker crosses two strips of wood at E, with BE = ED. AE = (x + 4) cm and EC = (2x − 4) cm. What must x be for the four ends to form a parallelogram?
Working:
Answer:
15.A four-sided bracket ABCD has sides, in order, AB = 23 cm, BC = 23 cm, CD = 14 cm and DA = 14 cm. It must be a parallelogram.
True or false?
Working:
Answer:
16.In ABCD, AB = (2x + 4) cm, DC = (x + 6) cm, AD = (y + 12) cm and BC = (3y − 4) cm. For both pairs of opposite sides to be equal, x = and y = .
Fill in each blank
Working:
Answer:
17.A four-sided bracket ABCD has sides, in order, AB = 41 cm, BC = 33 cm, CD = 41 cm and DA = 33 cm. It must be a parallelogram.
True or false?
Working:
Answer:
18.A worker crosses two strips of wood at E, with BE = ED. AE = (x + 2) cm and EC = (2x − 1) cm. What must x be for the four ends to form a parallelogram?
Working:
Answer:

Show the answers and working

Conditions for a parallelogram worksheet: Answers and working

1. 9
AB is already parallel to DC, so it needs AB = DC: 3x + 7 = 6x − 20, x = 9.
2. 5
AB is already parallel to DC, so it needs AB = DC: 2x + 3 = 4x − 7, x = 5.
3. 9
AB is already parallel to DC, so it needs AB = DC: 3x + 7 = 5x − 11, x = 9.
4. 7
BD is already halved; it needs AE = EC too (diags. bisect each other): x + 1 = 2x − 6, x = 7. Check: AE = EC = 8 cm.
5. 5
BD is already halved; it needs AE = EC too (diags. bisect each other): x + 4 = 2x − 1, x = 5. Check: AE = EC = 9 cm.

6. D. opp. sides equal
AB = DC (one tick) and AD = BC (two): both pairs of opposite sides equal.
7. True
∠D = 360° − 100° − 80° − 100°; only both pairs of opposite angles equal (∠A = ∠C and ∠B = ∠D) make it a parallelogram.
8. False
∠D = 360° − 105° − 75° − 120°; only both pairs of opposite angles equal (∠A = ∠C and ∠B = ∠D) make it a parallelogram.
9. False
∠D = 360° − 85° − 95° − 90°; only both pairs of opposite angles equal (∠A = ∠C and ∠B = ∠D) make it a parallelogram.
10. C. diags. bisect each other
AE = EC and BE = ED: the diagonals bisect each other.
11. 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.
12. D. not necessarily: it can be an isosceles trapezium
One pair parallel and the other pair equal can be an isosceles trapezium (here AB ≠ DC), so not necessarily a parallelogram.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. B. opp. sides equal
Both pairs of opposite sides are equal (61 cm and 41 cm): opp. sides equal.
14. 8
The diagonals must bisect each other, AE = EC: x + 4 = 2x − 4, so x = 8 and AE = EC = 12 cm.
15. False
Both pairs of opposite sides (AB with CD, BC with DA) must match; equal sides next to each other make a kite instead.
16. 2, 8
2x + 4 = x + 6, so x = 2; y + 12 = 3y − 4, so y = 8.
17. True
Both pairs of opposite sides (AB with CD, BC with DA) must match; equal sides next to each other make a kite instead.
18. 3
The diagonals must bisect each other, AE = EC: x + 2 = 2x − 1, so x = 3 and AE = EC = 5 cm.

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