Similar triangles · Ages 14–15

Proving triangles similar

One pair a line, with its reason; common ∠, vert. opp. ∠s, parallels.

How to do it

Method

One pair a line, with its reason; common ∠, vert. opp. ∠s, parallels

∠A = ∠A (common ∠), ∠ADE = ∠ABC (corr. ∠s), △ADE ~ △ABC (AAA)

Another way

By two sides and the angle between

ADAB = 39 = 13, AEAC = 412 = 13, ∠A common

In full

Written as a congruence proof is: start "In △ADE and △ABC,", then a pair of equal angles (or of proportional sides) a line, the reason in brackets, and last the conclusion with its condition. ∠A = ∠A (common ∠); ∠ADE = ∠ABC (corr. ∠s, DE // BC); so △ADE ~ △ABC (AAA). Reasons often used: common ∠, vert. opp. ∠s, corresponding and alternate angles of parallels, given, ∠ sum of △. With sides, write the ratios out: ADAB = 39 = 13, AEAC = 412 = 13, ∠A = ∠A (common ∠), so △ADE ~ △ABC (ratio of 2 sides, inc. ∠). Once similar, the triangles' corresponding sides give lengths (corr. sides, ~△s). Another way: work back from the end. Find two triangles holding the side you want, then look for their equal angles.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
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. common ∠
- B. alt. ∠s, AB // DC
- C. corr. ∠s, AB // DC
- D. vert. opp. ∠s
Choose the right answer
Show the answer and working
Answer: B. 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.
Example 2
In the picture, ∠ADE = ∠ACB, AB = 7 cm and AC = 6 cm. x =
Show the answer and working
Answer: 3 cm
△ADE ~ △ACB (AAA): ADAC = AEAB, so x = 6 × 3.5 ÷ 7 = 3.
Example 3
In the picture, ∠BAD = ∠BCA and BC = 10 cm. x = , y =
Fill in each blank
Show the answer and working
Answer: 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.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.In the picture, DE // BC. One line of the proof that △ADE ~ △ABC is: ∠DAE = ∠BAC ( ? ). Which reason goes in the gap?
- A. corr. ∠s, ~△s
- B. given
- C. vert. opp. ∠s
- D. common ∠
Choose the right answer
Answer:
2.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. vert. opp. ∠s
- B. corr. ∠s, ~△s
- C. corr. ∠s, DE // BC
- D. alt. ∠s, DE // BC
Choose the right answer
Answer:
3.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. corr. ∠s, AB // DC
- B. common ∠
- C. vert. opp. ∠s
- D. alt. ∠s, AB // DC
Choose the right answer
Answer:
4.In the picture, ∠ADE = ∠ACB, AB = 7 cm and AC = 6 cm. x =
Answer:
5.In the picture, ∠BAD = ∠BCA and BC = 10 cm. x = , y =
Fill in each blank
Answer:

All the answers

1. D. common ∠
2. C. corr. ∠s, DE // BC
3. D. alt. ∠s, AB // DC
4. 2.4 cm
5. 6.4, 3.6

Free worksheet: Proving triangles similarWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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