Similar triangles · Ages 14–15
Conditions for similar triangles
AAA, 3 sides proportional, 2 sides proportional with the included angle.
How to do it
Method
AAA, 3 sides proportional, 2 sides proportional with the included angle
64 = 96 = 128 = 1.5: 3 sides proportional
Another way
Find the third angle first
180° − 50° − 70° = 60°: both are 50°, 60°, 70°, AAA
In full
Any one of three conditions proves two triangles similar: AAA (the three angles equal; two equal angles force the third to be equal too, so finding two pairs is enough); 3 sides proportional (all three pairs of corresponding sides in the same ratio); and 2 sides proportional with the included angle equal (two pairs of sides in one ratio, and the angle between them equal). One way: compare sides. Sort each triangle's sides from short to long and divide pair by pair: 4, 6, 8 against 6, 9, 12 gives 64 = 96 = 128 = 1.5, so the sides are proportional. Another: compare angles. One triangle has 50° and 70°, so its third angle is 180° − 50° − 70° = 60°; the other has 60° and 70°, so its third is 50°: AAA. Careful: two sides in proportion with an angle that isn't the one between them prove nothing; and sides that each differ by the same amount (3, 4, 5 against 5, 6, 7) aren't similar: similar means every side times one number.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
In △ABC, AB = 8 cm, AC = 6 cm and ∠A = 60°. In △PQR, PQ = 12 cm and ∠P = 60°. For △ABC ~ △PQR (ratio of 2 sides, inc. ∠), PR =
Show the answer and working
Answer: 9 cm
∠A = ∠P lies between the sides, so the two ratios must match: PQAB = 128 = 1.5, so PR = 6 × 1.5 = 9 cm.
Example 2
A triangle with sides 4 cm, 7 cm and x cm is similar to one with sides 8 cm, 14 cm and 18 cm, x being the longest side. x =
Show the answer and working
Answer: 9 cm
Pair them short to long: 84 = 147 = 2, and x goes with 18: x = 18 ÷ 2 = 9.
Example 3
In △ABC and △PQR, PQAB = PRAC = 2. Which further fact proves △ABC ~ △PQR?
- A. ∠B = ∠Q
- B. ∠A = ∠P
- C. ∠C = ∠R
- D. BC = QR
Choose the right answer
Show the answer and working
Answer: B. ∠A = ∠P
∠A and ∠P lie between the proportional sides: ratio of 2 sides, inc. ∠.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.△ABC ~ △DEF, ∠A = 35° and ∠B = 70°. ∠F =
Answer:
2.The two triangles in the picture are similar. Which reason fits best?
- A. AAA
- B. ratio of 2 sides, inc. ∠
- C. 3 sides prop.
- D. ASA
Choose the right answer
Answer:
3.A triangle with sides 5 cm, 6 cm and 8 cm is similar to one with sides 7 cm, 8 cm and 10 cm.
True or false?
Answer:
4.In △ABC, AB = 3 cm, AC = 8 cm and ∠A = 50°. In △PQR, PQ = 9 cm and ∠P = 50°. For △ABC ~ △PQR (ratio of 2 sides, inc. ∠), PR =
Answer:
5.In △ABC and △PQR, PQAB = PRAC = 2.5. Which further fact proves △ABC ~ △PQR?
- A. BC = QR
- B. ∠C = ∠R
- C. ∠B = ∠Q
- D. ∠A = ∠P
Choose the right answer
Answer:
All the answers
- 1. 75°
- 2. A. AAA
- 3. False
- 4. 24 cm
- 5. D. ∠A = ∠P
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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