Geometry and proof · Ages 14–15
Conditions for similar triangles worksheet
Part of: Similar triangles
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Conditions for similar triangles worksheet
Geometry and proof · Ages 14–15
Numsprite
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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.△ABC ~ △DEF, ∠A = 30° and ∠B = 45°. ∠F =
Answer:
2.△ABC ~ △DEF, ∠A = 35° and ∠B = 80°. ∠F =
Answer:
3.In △ABC, AB = 6 cm, AC = 3 cm and ∠A = 40°. In △PQR, PQ = 12 cm and ∠P = 40°. For △ABC ~ △PQR (ratio of 2 sides, inc. ∠), PR =
Answer:
4.In △ABC, AB = 7 cm, AC = 3 cm and ∠A = 110°. In △PQR, PQ = 21 cm and ∠P = 110°. For △ABC ~ △PQR (ratio of 2 sides, inc. ∠), PR =
Answer:
5.A triangle with sides 5 cm, 6 cm and x cm is similar to one with sides 12.5 cm, 15 cm and 17.5 cm, x being the longest side. x =
Answer:
Part B: Choose, or true or false (2 marks each)
6.A triangle with sides 5 cm, 7 cm and 9 cm is similar to one with sides 6 cm, 8 cm and 10 cm.
True or false?
Answer:
7.The two triangles in the picture are similar. Which reason fits best?
- A. ASA
- B. 3 sides prop.
- C. AAA
- D. ratio of 2 sides, inc. ∠
Choose the right answer
Answer:
8.A triangle with sides 3 cm, 6 cm and 8 cm is similar to one with sides 4 cm, 7 cm and 9 cm.
True or false?
Answer:
9.A triangle has sides 6 cm, 7 cm and 10 cm. Which of these triangles is similar to it?
- A. 8 cm, 9 cm, 12 cm
- B. 12 cm, 14 cm, 10 cm
- C. 12 cm, 14 cm, 20 cm
- D. 14 cm, 14 cm, 20 cm
Choose the right answer
Answer:
10.A triangle has sides 5 cm, 6 cm and 7 cm. Which of these triangles is similar to it?
- A. 12.5 cm, 15 cm, 17.5 cm
- B. 15 cm, 15 cm, 17.5 cm
- C. 12.5 cm, 15 cm, 7 cm
- D. 7 cm, 8 cm, 9 cm
Choose the right answer
Answer:
11.In △ABC and △PQR, PQAB = PRAC = 2. Which further fact proves △ABC ~ △PQR?
- A. ∠C = ∠R
- B. BC = QR
- C. ∠B = ∠Q
- D. ∠A = ∠P
Choose the right answer
Answer:
12.In △ABC and △PQR, PQAB = PRAC = 2.5. Which further fact proves △ABC ~ △PQR?
- A. ∠B = ∠Q
- B. ∠A = ∠P
- C. BC = QR
- D. ∠C = ∠R
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.Two right-angled wooden wedges: the small one's legs are 9 cm and 12 cm and its hypotenuse 15 cm; the big one's legs are 27 cm and 36 cm. The legs are in proportion and the angle between them (a right angle) is equal, so they are similar. Scale = , and the big wedge's hypotenuse = cm.
Fill in each blank
Working:
Answer:
14.Two triangular flower beds have the same shape. Two angles of the first are 70° and 80°. What is the largest angle of the second?
Working:
Answer:
15.A triangular stand is made of rods 4 dm, 8 dm and 9 dm long. To make a bigger stand of the same shape, which three rods should be used?
- A. 5 dm, 9 dm, 10 dm
- B. 8 dm, 8 dm, 18 dm
- C. 8 dm, 16 dm, 18 dm
- D. 8 dm, 16 dm, 10 dm
Choose the right answer
Working:
Answer:
16.A worker checks whether two triangular frames are similar. The small one's sides are 3 cm, 7 cm and 8 cm; the big one's 4.5 cm, 10.5 cm and 13 cm. Dividing pair by pair, short to long: , , . (The frames are similar only if all three are equal.)
Fill in each blank
Working:
Answer:
17.A worker checks whether two triangular frames are similar. The small one's sides are 4 cm, 8 cm and 10 cm; the big one's 6 cm, 12 cm and 16 cm. Dividing pair by pair, short to long: , , . (The frames are similar only if all three are equal.)
Fill in each blank
Working:
Answer:
18.A triangular stand is made of rods 5 dm, 7 dm and 8 dm long. To make a bigger stand of the same shape, which three rods should be used?
- A. 6 dm, 8 dm, 9 dm
- B. 15 dm, 15 dm, 24 dm
- C. 15 dm, 21 dm, 24 dm
- D. 15 dm, 21 dm, 9 dm
Choose the right answer
Working:
Answer:
Show the answers and working
Conditions for similar triangles worksheet: Answers and working
Part A: Working out and filling in
1. 105°
∠F goes with ∠C = 180° − 30° − 45° = 105° (∠ sum of △; corr. ∠s, ~△s).
2. 65°
∠F goes with ∠C = 180° − 35° − 80° = 65° (∠ sum of △; corr. ∠s, ~△s).
3. 6 cm
∠A = ∠P lies between the sides, so the two ratios must match: PQAB = 126 = 2, so PR = 3 × 2 = 6 cm.
4. 9 cm
∠A = ∠P lies between the sides, so the two ratios must match: PQAB = 217 = 3, so PR = 3 × 3 = 9 cm.
5. 7 cm
Pair them short to long: 12.55 = 156 = 2.5, and x goes with 17.5: x = 17.5 ÷ 2.5 = 7.
Part B: Choose, or true or false
6. False
Divide pair by pair, short to long: 65, 87, 109. Only three equal ratios make the sides proportional; adding the same number to each side doesn't.
7. C. AAA
Two pairs of equal angles (75° and 50°) force the third pair to be equal too (∠ sum of △), so AAA.
8. False
Divide pair by pair, short to long: 43, 76, 98. Only three equal ratios make the sides proportional; adding the same number to each side doesn't.
9. C. 12 cm, 14 cm, 20 cm
126 = 147 = 2010 = 2: 3 sides proportional.
10. A. 12.5 cm, 15 cm, 17.5 cm
12.55 = 156 = 17.57 = 2.5: 3 sides proportional.
11. D. ∠A = ∠P
∠A and ∠P lie between the proportional sides: ratio of 2 sides, inc. ∠.
12. B. ∠A = ∠P
∠A and ∠P lie between the proportional sides: ratio of 2 sides, inc. ∠.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 3, 45
27 ÷ 9 = 36 ÷ 12 = 3 (ratio of 2 sides, inc. ∠); hypotenuse = 15 × 3 = 45 cm.
14. 80°
The third angle = 180° − 70° − 80°; similar triangles have equal angles, so the second has the same three, the largest 80°.
15. C. 8 dm, 16 dm, 18 dm
Every rod times 2: 84 = 168 = 189 = 2, 3 sides proportional.
16. 1.5, 1.5, 1.625
4.5 ÷ 3 = 1.5, 10.5 ÷ 7 = 1.5, 13 ÷ 8 = 1.625; only three equal ratios make the sides proportional.
17. 1.5, 1.5, 1.6
6 ÷ 4 = 1.5, 12 ÷ 8 = 1.5, 16 ÷ 10 = 1.6; only three equal ratios make the sides proportional.
18. C. 15 dm, 21 dm, 24 dm
Every rod times 3: 155 = 217 = 248 = 3, 3 sides proportional.
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