Geometry and proof · Ages 14–15

Congruent quadrilaterals 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.The parallelogram ABCD ≅ the parallelogram PQRS, and ∠A = 85°. ∠R =
Answer:
2.Two rhombuses both have 4 cm sides. One of them has an angle of 55°. For them to be congruent, the other's largest angle =
Answer:
3.The parallelogram ABCD ≅ the parallelogram PQRS, and ∠A = 50°. ∠R =
Answer:
4.Two rhombuses both have 3 cm sides. One of them has an angle of 130°. For them to be congruent, the other's largest angle =
Answer:
5.ABCD ≅ PQRS, with ∠P = 110°, ∠B = 80° and ∠R = 65°. ∠D =
Answer:
6.ABCD ≅ PQRS, with ∠P = 75°, ∠B = 120° and ∠R = 65°. ∠D =
Answer:

7.Which pair of quadrilaterals must be congruent?
- A. two rectangles each with a perimeter of 18 cm
- B. two squares with 6 cm sides
- C. two rectangles both 6 cm long
- D. two rhombuses with 6 cm sides
Choose the right answer
Answer:
8.A square with 3 cm sides and a square with 3 cm sides are congruent.
True or false?
Answer:
9.A square with 11 cm sides and a square with 11 cm sides are congruent.
True or false?
Answer:
10.ABCD ≅ PQRS, with AB = 3 cm, BC = 4 cm, CD = 8 cm and DA = 6 cm. How long is RS?
- A. 4 cm
- B. 3 cm
- C. 8 cm
- D. 6 cm
Choose the right answer
Answer:
11.ABCD ≅ PQRS, with AB = 8 cm, BC = 5 cm, CD = 3 cm and DA = 7 cm. How long is RS?
- A. 3 cm
- B. 8 cm
- C. 7 cm
- D. 5 cm
Choose the right answer
Answer:
12.Which set of matching parts being equal is enough to make two quadrilaterals congruent?
- A. two diagonals
- B. four sides
- C. four sides and a diagonal
- D. four angles
Choose the right answer
Answer:

13.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 11 cm
- B. all four sides 11 cm
- C. all four sides 11 cm, and one matching angle
- D. all four angles 90°
Choose the right answer
Working:
Answer:
14.Two rectangular cards: one 15 cm by 7 cm, the other 15 cm by 7 cm. The two cards are congruent.
True or false?
Working:
Answer:
15.Two parallelogram cards ABCD ≅ PQRS, with AB = 12 cm, BC = 9 cm and ∠A = 75°. RS = cm, SP = cm, ∠Q = °.
Fill in each blank
Working:
Answer:
16.Two kites are cut from one pattern, so they are congruent. The first, ABCD, has ∠A = 120° and ∠B = ∠D = 75°. The second, PQRS ≅ ABCD. What is ∠R?
Working:
Answer:
17.Two kites are cut from one pattern, so they are congruent. The first, ABCD, has ∠A = 70° and ∠B = ∠D = 105°. The second, PQRS ≅ ABCD. What is ∠R?
Working:
Answer:
18.Two parallelogram cards ABCD ≅ PQRS, with AB = 7 cm, BC = 4 cm and ∠A = 55°. RS = cm, SP = cm, ∠Q = °.
Fill in each blank
Working:
Answer:

Show the answers and working

Congruent quadrilaterals worksheet: Answers and working

1. 85°
∠R goes with ∠C, and ∠C = ∠A = 85° (opp. ∠s of // gram), so ∠R = 85°.
2. 125°
The first rhombus has angles 55° and 180° − 55° (int. ∠s); congruent means the same angles, the largest 125°.
3. 50°
∠R goes with ∠C, and ∠C = ∠A = 50° (opp. ∠s of // gram), so ∠R = 50°.
4. 130°
The first rhombus has angles 130° and 180° − 130° (int. ∠s); congruent means the same angles, the largest 130°.
5. 105°
∠A = ∠P = 110° and ∠C = ∠R = 65° (matching angles); ∠D = 360° − 110° − 80° − 65° = 105° (∠ sum of polygon).
6. 100°
∠A = ∠P = 75° and ∠C = ∠R = 65° (matching angles); ∠D = 360° − 75° − 120° − 65° = 100° (∠ sum of polygon).

7. B. two squares with 6 cm sides
A square's angles are fixed at 90°, and the sides match too, so they fit exactly.
8. True
A square's angles are all 90°; two squares with the same side fit exactly, with different sides they don't.
9. True
A square's angles are all 90°; two squares with the same side fit exactly, with different sides they don't.
10. C. 8 cm
In order: R goes with C and S with D, so RS = CD = 8 cm.
11. A. 3 cm
In order: R goes with C and S with D, so RS = CD = 3 cm.
12. C. four sides and a diagonal
A diagonal cuts each quadrilateral into two triangles, each pair with three equal sides (SSS), so the whole shapes are congruent.

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

13. C. all four sides 11 cm, and one matching angle
Four equal sides make a rhombus; the same side and one same angle make two rhombuses congruent.
14. True
A rectangle's angles are all 90°; it takes the same length and width, not just the same perimeter.
15. 12, 9, 105
RS = CD = AB = 12 cm and SP = DA = BC = 9 cm; ∠Q = ∠B = 180° − 75° = 105°.
16. 90°
∠R goes with ∠C = 360° − 120° − 2 × 75° = 90° (∠ sum of polygon).
17. 80°
∠R goes with ∠C = 360° − 70° − 2 × 105° = 80° (∠ sum of polygon).
18. 7, 4, 125
RS = CD = AB = 7 cm and SP = DA = BC = 4 cm; ∠Q = ∠B = 180° − 55° = 125°.

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