Geometry and proof · Ages 14–15
Quadrilaterals worksheet
Covers: Properties of a parallelogram, Rectangles and squares, Rhombuses, Conditions for a parallelogram, Proofs with quadrilaterals, The mid-point theorem, The intercept theorem, Congruent quadrilaterals
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Quadrilaterals worksheet
Geometry and proof · Ages 14–15
Numsprite
numsprite.com
Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.ABCD is a parallelogram. x =
Answer:
2.ABCD is a rectangle whose diagonals cross at E. ∠EBA = °, ∠AEB = °, ∠EBC = °
Fill in each blank
Answer:
3.ABCD is a rectangle whose diagonals cross at E. x =
Answer:
4.A rhombus has diagonals of 48 cm and 36 cm. Its side =
Answer:
5.ABCD is a rhombus whose diagonals cross at E. x =
Answer:
6.In the parallelogram ABCD, the bisector of ∠DAB meets DC at F, with AD = 4 cm and AB = 15 cm. FC =
Answer:
7.In the picture, M is the mid-point of AB and N the mid-point of AC. x =
Answer:
Part B: Choose, or true or false (2 marks each)
8.Which of these is not always a property of a parallelogram?
- A. opposite angles are equal
- B. opposite sides are equal
- C. the diagonals are equal
- D. the diagonals bisect each other
Choose the right answer
Answer:
9.By the marks in the picture, ABCD is a parallelogram. Why?
- A. diags. bisect each other
- B. opp. sides equal and //
- C. opp. sides equal
- D. opp. ∠s equal
Choose the right answer
Answer:
10.By the marks in the picture, ABCD is a parallelogram. Why?
- A. opp. ∠s equal
- B. opp. sides equal and //
- C. diags. bisect each other
- D. opp. sides equal
Choose the right answer
Answer:
11.Three parallel lines cut two lines at A, B, C and at D, E, F. AB = 5 cm and BC = 6 cm, so DE = EF.
True or false?
Answer:
12.ABCD ≅ PQRS, with AB = 6 cm, BC = 9 cm, CD = 7 cm and DA = 8 cm. How long is RS?
- A. 8 cm
- B. 6 cm
- C. 7 cm
- D. 9 cm
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.A floor tile is a parallelogram, one of its angles 55°. With ∠A = 55°, the other three are ∠B = °, ∠C = ° and ∠D = °.
Fill in each blank
Working:
Answer:
14.A rhombus kite's two struts (its diagonals) are 32 dm and 24 dm, crossing at right angles at their middles. Each side is dm, and the string round the frame is dm long.
Fill in each blank
Working:
Answer:
15.A worker crosses two strips of wood at E, with BE = ED. AE = (x + 5) cm and EC = (2x − 4) cm. What must x be for the four ends to form a parallelogram?
Working:
Answer:
16.A stepladder's frame is a trapezium 32 cm wide at the bottom and 20 cm at the top; a bar joins the mid-points of the two sides, parallel to the bottom. How long is the bar?
Working:
Answer:
17.A sheet has parallel, evenly spaced ruled lines. A strip 28 cm long is laid slantwise with its ends on two lines, spanning 7 gaps. How long is each piece the lines cut it into?
Working:
Answer:
18.A factory turns out four-sided tiles that must all be the same. Which set of matching measurements guarantees two tiles are congruent?
- A. one diagonal 5 cm
- B. all four sides 5 cm, and one matching angle
- C. all four angles 90°
- D. all four sides 5 cm
Choose the right answer
Working:
Answer:
Show the answers and working
Quadrilaterals worksheet: Answers and working
Part A: Working out and filling in
1. 100°
AD // BC, so ∠A and ∠B are interior angles: x = 180° − 80° = 100°.
2. 30, 120, 60
EA = EB, so ∠EBA = 30°; ∠AEB = 180° − 2 × 30° = 120°; ∠EBC = 90° − 30° = 60°.
3. 88°
EA = EB (prop. of rectangle), so ∠EBA = 46° (base ∠s, isos. △); x = 180° − 2 × 46° = 88°.
4. 30 cm
The diagonals bisect each other at right angles, so each right triangle has legs 48 ÷ 2 and 36 ÷ 2: side = (p2)2 + (q2)2 = 30 cm (Pyth. thm.).
5. 39°
∠AEB = 90° (prop. of rhombus); x = 180° − 90° − 51° = 39° (∠ sum of △).
6. 11 cm
∠DFA = ∠FAB (alt. ∠s, AB // DC) = ∠DAF, so DF = DA = 4 cm (sides opp. equal ∠s); DC = AB = 15 cm, so FC = 15 − 4 = 11 cm.
7. 4 cm
Mid-point theorem: MN = BC ÷ 2, so x = 4.
Part B: Choose, or true or false
8. C. the diagonals are equal
Equal diagonals belong to rectangles; a general parallelogram's diagonals differ in length.
9. C. opp. sides equal
AB = DC (one tick) and AD = BC (two): both pairs of opposite sides equal.
10. B. 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.
11. False
The intercept theorem needs AB = BC to give DE = EF; 5 cm and 6 cm must be equal.
12. C. 7 cm
In order: R goes with C and S with D, so RS = CD = 7 cm.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 125, 55, 125
∠B = 180° − 55° = 125° (int. ∠s); ∠C = ∠A = 55° and ∠D = ∠B = 125° (opp. ∠s of // gram).
14. 20, 80
A rhombus's diagonals bisect each other at right angles: side = (32 ÷ 2)2 + (24 ÷ 2)2 = 20 dm; perimeter = 4 × 20 = 80 dm.
15. 9
The diagonals must bisect each other, AE = EC: x + 5 = 2x − 4, so x = 9 and AE = EC = 14 cm.
16. 26 cm
Draw a diagonal and use the mid-point theorem twice: 32 ÷ 2 + 20 ÷ 2 = (32 + 20) ÷ 2 = 26 cm.
17. 4 cm
Evenly spaced parallels cut equal pieces off the strip too (intercept thm.): 28 ÷ 7 = 4 cm.
18. B. all four sides 5 cm, and one matching angle
Four equal sides make a rhombus; the same side and one same angle make two rhombuses congruent.
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