Geometry and proof · Ages 14–15
Proving triangles similar worksheet
Part of: Similar triangles
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Proving triangles similar 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.In the picture, DE // BC. Find DE: put the steps in order.
- A. DE = 8 × 38 = 3 cm
- B. ∠A = ∠A (common ∠), ∠ADE = ∠ABC (corr. ∠s, DE // BC), so △ADE ~ △ABC (AAA)
- C. DEBC = ADAB (corr. sides, ~△s)
Write the letters in the right order
Answer:
2.In the picture, ∠ADE = ∠ACB, AB = 9 cm and AC = 11 cm. x =
Answer:
3.In the picture, ∠BAD = ∠BCA and BC = 10 cm. x = , y =
Fill in each blank
Answer:
4.In the picture, ∠ADE = ∠ACB, AB = 11 cm and AC = 12 cm. x =
Answer:
5.In the picture, ∠ADE = ∠ACB, AB = 10 cm and AC = 11 cm. x =
Answer:
6.In the picture, ∠BAD = ∠BCA and BC = 8 cm. x = , y =
Fill in each blank
Answer:
7.In the picture, ∠ADE = ∠ACB, AB = 9 cm and AC = 10 cm. x =
Answer:
8.In the picture, ∠ADE = ∠ACB, AB = 12 cm and AC = 11 cm. x =
Answer:
Part B: Choose, or true or false (2 marks each)
9.In the picture, DE // BC. One line of the proof that △ADE ~ △ABC is: ∠DAE = ∠BAC ( ? ). Which reason goes in the gap?
- A. given
- B. common ∠
- C. corr. ∠s, ~△s
- D. vert. opp. ∠s
Choose the right answer
Answer:
10.In the picture, DE // BC. In △ADE and △ABC: ∠A = ∠A (common ∠), ∠ADE = ∠ABC ( ? ), so △ADE ~ △ABC (AAA). Which reason goes in the gap?
- A. alt. ∠s, DE // BC
- B. vert. opp. ∠s
- C. corr. ∠s, DE // BC
- D. corr. ∠s, ~△s
Choose the right answer
Answer:
11.In the picture, AB // DC, and AC and BD cross at E. In △ABE and △CDE: ∠BAE = ∠DCE ( ? ). Which reason goes in the gap?
- A. vert. opp. ∠s
- B. common ∠
- C. alt. ∠s, AB // DC
- D. corr. ∠s, AB // DC
Choose the right answer
Answer:
12.In the picture, ∠ADE = ∠ACB. Which triangle is △ADE similar to, its vertices in matching order?
- A. △BCA
- B. △CAB
- C. △ABC
- D. △ACB
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.In △ABC, D is on AB and E on AC, with AD = 2 cm, AB = 8 cm, AE = 6 cm and AC = 24 cm. Prove DE // BC: put the steps in order.
- A. ADAB = 28, AEAC = 624 = 28
- B. ∠ADE = ∠ABC (corr. ∠s, ~△s), so DE // BC (corr. ∠s equal)
- C. ∠A = ∠A (common ∠), so △ADE ~ △ABC (ratio of 2 sides, inc. ∠)
Write the letters in the right order
Working:
Answer:
14.A triangular board ABC has D on BC with ∠BAD = ∠BCA. BD = 27 dm and DC = 21 dm. △BAD ~ △BCA (∠B common, AAA), so BDBA = BABC. BC = dm, AB = dm.
Fill in each blank
Working:
Answer:
15.In the picture, AB // DC, and AC and BD cross at E. Prove ABCD = AECE: put the steps in order.
- A. ∠BAE = ∠DCE (alt. ∠s, AB // DC)
- B. ABCD = AECE (corr. sides, ~△s)
- C. ∠AEB = ∠CED (vert. opp. ∠s), so △ABE ~ △CDE (AAA)
Write the letters in the right order
Working:
Answer:
16.In the picture, DE // BC. In △ADE and △ABC: ∠ADE = ∠ABC (corr. ∠s, DE // BC), ∠AED = ∠ACB (corr. ∠s, DE // BC), ∠A = ∠A (common ∠), so △ADE ~ △ABC ( ? ). Which condition goes in the gap?
- A. 3 sides prop.
- B. ratio of 2 sides, inc. ∠
- C. AAA
- D. ASA
Choose the right answer
Working:
Answer:
17.A triangular board ABC has D on BC with ∠BAD = ∠BCA. BD = 18 dm and DC = 14 dm. △BAD ~ △BCA (∠B common, AAA), so BDBA = BABC. BC = dm, AB = dm.
Fill in each blank
Working:
Answer:
18.In △ABC, D is on AB and E on AC, with AD = 2 cm, AB = 5 cm, AE = 4 cm and AC = 10 cm. Prove DE // BC: put the steps in order.
- A. ∠ADE = ∠ABC (corr. ∠s, ~△s), so DE // BC (corr. ∠s equal)
- B. ADAB = 25, AEAC = 410 = 25
- C. ∠A = ∠A (common ∠), so △ADE ~ △ABC (ratio of 2 sides, inc. ∠)
Write the letters in the right order
Working:
Answer:
Show the answers and working
Proving triangles similar worksheet: Answers and working
Part A: Working out and filling in
1. B → C → A
Similarity first; only then do the corresponding sides give the proportion, and last AB = 8 cm goes in.
2. 6.6 cm
△ADE ~ △ACB (AAA): ADAC = AEAB, so x = 11 × 5.4 ÷ 9 = 6.6.
3. 4.9, 5.1
∠B is common, so △BAD ~ △BCA (AAA): BDBA = BABC, x = 7 × 7 ÷ 10 = 4.9, and y = 10 − 4.9 = 5.1.
4. 7.2 cm
△ADE ~ △ACB (AAA): ADAC = AEAB, so x = 12 × 6.6 ÷ 11 = 7.2.
5. 6.6 cm
△ADE ~ △ACB (AAA): ADAC = AEAB, so x = 11 × 6 ÷ 10 = 6.6.
6. 3.125, 4.875
∠B is common, so △BAD ~ △BCA (AAA): BDBA = BABC, x = 5 × 5 ÷ 8 = 3.125, and y = 8 − 3.125 = 4.875.
7. 5 cm
△ADE ~ △ACB (AAA): ADAC = AEAB, so x = 10 × 4.5 ÷ 9 = 5.
8. 6.6 cm
△ADE ~ △ACB (AAA): ADAC = AEAB, so x = 11 × 7.2 ÷ 12 = 6.6.
Part B: Choose, or true or false
9. B. common ∠
∠DAE and ∠BAC are the same angle, shared by both triangles: common ∠.
10. C. corr. ∠s, DE // BC
AB cuts the parallels DE and BC, and ∠ADE and ∠ABC sit in the same position: corr. ∠s, DE // BC.
11. C. alt. ∠s, AB // DC
AC cuts the parallels AB and DC, and ∠BAE and ∠DCE are on either side of it: alt. ∠s, AB // DC.
12. D. △ACB
∠A is common and ∠ADE = ∠ACB, so A goes with A, D with C, E with B: △ADE ~ △ACB (AAA).
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. A → C → B
Compare the ratios and the angle between to prove similarity; similarity gives equal corresponding angles, and equal corresponding angles make the lines parallel.
14. 48, 36
BC = 27 + 21 = 48; AB × AB = BD × BC = 27 × 48, so AB = 36 dm.
15. A → C → B
Two pairs of equal angles first (alternate, vertically opposite), then similarity, and only then the corresponding sides.
16. C. AAA
Three pairs of equal angles: AAA.
17. 32, 24
BC = 18 + 14 = 32; AB × AB = BD × BC = 18 × 32, so AB = 24 dm.
18. B → C → A
Compare the ratios and the angle between to prove similarity; similarity gives equal corresponding angles, and equal corresponding angles make the lines parallel.
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