Geometry and proof · Ages 13–14
Conditions for congruent triangles worksheet
Part of: Congruent triangles
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Conditions for congruent triangles worksheet
Geometry and proof · Ages 13–14
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.Two triangles have two pairs of equal angles, 46° and 64°. Their third angles are both
Answer:
2.Two triangles have two pairs of equal angles, 90° and 47°. Their third angles are both
Answer:
3.Two triangles have two pairs of equal angles, 82° and 77°. Their third angles are both
Answer:
Part B: Choose, or true or false (2 marks each)
4.By the marks (parts with the same marks are equal), which condition gives △ABC ≅ △PQR?
- A. ASA
- B. SAS
- C. SSS
- D. AAS
Choose the right answer
Answer:
5.By the marks (parts with the same marks are equal; a square is a right angle), △ABC and △PQR must be congruent.
True or false?
Answer:
6.In △ABC and △PQR, AB = PQ, ∠A = ∠P and ∠B = ∠Q. Which condition makes them congruent?
- A. SAS
- B. SSS
- C. AAS
- D. ASA
Choose the right answer
Answer:
7.By the marks (parts with the same marks are equal; a square is a right angle), which condition gives △ABC ≅ △PQR?
- A. RHS
- B. SAS
- C. AAS
- D. SSS
Choose the right answer
Answer:
8.In the picture, AB = PQ and AC = PR. Which other fact proves △ABC ≅ △PQR by SAS?
- A. BC = QR
- B. ∠C = ∠R
- C. ∠B = ∠Q
- D. ∠A = ∠P
Choose the right answer
Answer:
9.In △ABC and △PQR, AB = PQ, ∠B = ∠Q and ∠C = ∠R. Which condition makes them congruent?
- A. SSS
- B. AAS
- C. SAS
- D. ASA
Choose the right answer
Answer:
10.In the picture, AB = PQ and ∠B = ∠Q. Which one more fact would still not prove △ABC ≅ △PQR?
- A. ∠C = ∠R
- B. AC = PR
- C. BC = QR
- D. ∠A = ∠P
Choose the right answer
Answer:
11.In △ABC, ∠A = 62°, ∠B = 49° and BC = 4 cm; in △PQR, ∠P = 62°, ∠R = 55° and QR = 4 cm. The two triangles must be congruent.
True or false?
Answer:
12.In △ABC, ∠A = 37°, ∠B = 52° and BC = 7 cm; in △PQR, ∠P = 37°, ∠R = 63° and QR = 7 cm. The two triangles must be congruent.
True or false?
Answer:
Part C: Word problems (4 marks each)
13.The equal sides and angles of two triangular metal plates are marked (a square is a right angle). Which condition makes them congruent?
- A. ASA
- B. AAS
- C. RHS
- D. SAS
Choose the right answer
Working:
Answer:
14.A pole AD stands upright on the ground BC, and two equal cables AB and AC run from its top to the ground. In △ABD and △ACD, ∠ADB = ∠ADC = 90°, AB = AC and AD = AD. Which condition makes them congruent?
- A. RHS
- B. ASA
- C. SAS
- D. SSS
Choose the right answer
Working:
Answer:
15.In △ABC and △PQR, AB = PQ, AC = PR and ∠A = ∠P. Which condition makes them congruent?
- A. They can't be shown congruent
- B. SAS
- C. SSS
- D. ASA
Choose the right answer
Working:
Answer:
16.A right-angled tile ABC has ∠B = 90°, AC = 13 cm, AB = 5 cm and BC = 12 cm. Another right-angled tile PQR has ∠Q = 90°, PR = 13 cm and PQ = 5 cm. QR =
Working:
Answer:
17.A right-angled tile ABC has ∠B = 90°, AC = 25 cm, AB = 7 cm and BC = 24 cm. Another right-angled tile PQR has ∠Q = 90°, PR = 25 cm and PQ = 7 cm. QR =
Working:
Answer:
18.A right-angled tile ABC has ∠B = 90°, AC = 20 cm, AB = 12 cm and BC = 16 cm. Another right-angled tile PQR has ∠Q = 90°, PR = 20 cm and PQ = 12 cm. QR =
Working:
Answer:
Show the answers and working
Conditions for congruent triangles worksheet: Answers and working
Part A: Working out and filling in
1. 70°
180° − 46° − 64° = 70° (∠ sum of △): two equal angles force the third to be equal, which is why AAS can be seen as ASA.
2. 43°
180° − 90° − 47° = 43° (∠ sum of △): two equal angles force the third to be equal, which is why AAS can be seen as ASA.
3. 21°
180° − 82° − 77° = 21° (∠ sum of △): two equal angles force the third to be equal, which is why AAS can be seen as ASA.
Part B: Choose, or true or false
4. C. SSS
SSS: three sides; SAS: two sides and the angle between; ASA: two angles and the side between; AAS: two angles and a side facing one of them.
5. True
SSS, SAS, ASA, AAS and RHS each prove congruence; three equal angles (AAA), or two sides and an angle not between them (SSA), are not enough: the triangles may differ.
6. D. ASA
The side between ∠A and ∠B is AB itself: two angles and the side between, ASA.
7. C. AAS
RHS: right angle, hypotenuse and a side; SAS: two sides and the angle between; SSS: three sides; AAS: two angles and a side facing one.
8. D. ∠A = ∠P
SAS needs the angle between the two sides: AB and AC meet at A, so ∠A = ∠P.
9. B. AAS
Two pairs of angles (∠B, ∠C) and a pair of sides not between them (AB faces ∠C): AAS.
10. B. AC = PR
AB, AC and ∠B: ∠B isn't between AB and AC (the angle between them is ∠A). That is SSA, which can make two different triangles, so it proves nothing.
11. False
Find the third angles first: ∠C = 180° − 62° − 49° = 69° and ∠Q = 180° − 62° − 55° = 63° (∠ sum of △). Only when the two triangles have the same three angles, with BC and QR both facing the 62° angle, does AAS prove them congruent; with different angles they can't be.
12. False
Find the third angles first: ∠C = 180° − 37° − 52° = 91° and ∠Q = 180° − 37° − 63° = 80° (∠ sum of △). Only when the two triangles have the same three angles, with BC and QR both facing the 37° angle, does AAS prove them congruent; with different angles they can't be.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. C. RHS
SAS: two sides and the angle between; ASA: two angles and the side between; AAS: two angles and a side facing one; RHS: right angle, hypotenuse and a side.
14. A. RHS
Both are right-angled, the hypotenuses AB = AC, and AD is a common side: RHS.
15. B. SAS
AB and AC meet at A, so ∠A is the angle between them: two sides and the angle between, SAS.
16. 12 cm
∠B = ∠Q = 90°, the hypotenuses AC = PR = 13 cm and AB = PQ = 5 cm, so △ABC ≅ △PQR (RHS) and QR = BC = 12 cm (corr. sides, ≅△s).
17. 24 cm
∠B = ∠Q = 90°, the hypotenuses AC = PR = 25 cm and AB = PQ = 7 cm, so △ABC ≅ △PQR (RHS) and QR = BC = 24 cm (corr. sides, ≅△s).
18. 16 cm
∠B = ∠Q = 90°, the hypotenuses AC = PR = 20 cm and AB = PQ = 12 cm, so △ABC ≅ △PQR (RHS) and QR = BC = 16 cm (corr. sides, ≅△s).
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