Similar triangle reasons and how to write the proof
The three reasons for similar triangles in the Hong Kong syllabus, how to set out the proof line by line, three worked examples and the RHS and SSA traps.
There are three reasons for similar triangles: AAA (or AA), 3 sides prop., and ratio of 2 sides, inc. ∠. Any one of them is enough to prove two triangles similar. Write the triangles in matching order, give every fact with its reason, and end with the condition in brackets.
Key points
- AAA (or AA): two pairs of corresponding angles are equal.
- 3 sides prop.: all three pairs of corresponding sides are in the same ratio.
- ratio of 2 sides, inc. ∠: two pairs of sides are in the same ratio and the angles between them are equal.
- RHS, SSS, SAS and SSA are not reasons for similar triangles.
The three conditions for similar triangles
The Education Bureau's junior secondary curriculum defines similar triangles as two triangles with all corresponding angles equal and all corresponding sides proportional. You do not have to check all six facts. The curriculum lists three conditions, and any one of them is enough to prove two triangles similar.
You are not asked to prove why these conditions work. You are asked to use them: to prove that two triangles are similar, then to find unknown sides and angles.
The curriculum writes the first condition as AAA (or AA). Two pairs of equal angles are enough, because the angle sum of a triangle fixes the third pair. Many teachers still expect AAA in a written proof, so use the form your teacher and textbook use.
The three conditions
| Condition | What you must show | Common abbreviation |
|---|---|---|
| Equal angles | Two pairs of corresponding angles are equal (the third pair follows) | AAA (or AA) |
| Corresponding sides proportional | All three ratios of corresponding sides are equal | 3 sides prop. |
| Two sides proportional, included angle equal | Two ratios of corresponding sides are equal, and the angles between those sides are equal | ratio of 2 sides, inc. ∠ |
Read next: Similar triangles: methods and worked examples
How to set out the proof
A marker looks for two things: a reason for every step, and a conclusion that names the condition. The order of the letters matters too, because the next part of the question usually asks for a length, and the order tells you which sides pair up.
- Mark the equal angles on the figure. If the two triangles overlap, sketch them separately, turned the same way round.
- Name the triangles in matching order. If ∠A matches ∠P and ∠B matches ∠Q, write △ABC ~ △PQR, never △ABC ~ △QPR.
- Write one fact per line, with its reason after it, such as (common ∠), (vert. opp. ∠s) or (corr. ∠s, AB // CD).
- Finish with the conclusion and the condition: ∴ △ABC ~ △PQR (AAA).
Read next: Prove similar: the method step by step
Example 1: two pairs of equal angles
In △ABC, P lies on AB and Q lies on AC, with PQ // BC. AP = 3 cm, PB = 2 cm and PQ = 4.5 cm. Prove that △APQ ~ △ABC, then find BC.
Now use the matching order. AB = 3 + 2 = 5 cm, and PQ/BC = AP/AB, so 4.5/BC = 3/5 and BC = 7.5 cm. Notice that AP is paired with the whole side AB, not with the piece PB.
Proof
| Statement | Reason |
|---|---|
| ∠APQ = ∠ABC | corr. ∠s, PQ // BC |
| ∠AQP = ∠ACB | corr. ∠s, PQ // BC |
| ∠PAQ = ∠BAC | common ∠ |
| ∴ △APQ ~ △ABC | AAA |
Read next: Find a length in similar triangles
Example 2: three sides in the same ratio
In △ABC, AB = 6 cm, BC = 8 cm and CA = 10 cm. In △DEF, DE = 9 cm, EF = 12 cm and FD = 15 cm. Prove that the triangles are similar.
Pair the shortest side with the shortest, the middle with the middle and the longest with the longest. The pairs of sides then tell you the matching vertices: A goes with D, B with E and C with F.
Proof
| Statement | Reason |
|---|---|
| DE/AB = 9/6 = 3/2 | |
| EF/BC = 12/8 = 3/2 | |
| FD/CA = 15/10 = 3/2 | |
| ∴ △ABC ~ △DEF | 3 sides prop. |
Example 3: two sides and the included angle
The straight lines AE and BD cross at C. CA = 4 cm, CE = 10 cm, CB = 6 cm, CD = 15 cm and AB = 5 cm. Prove that △CAB ~ △CED, then find ED.
The equal angle must sit between the two pairs of sides you used. ∠ACB lies between CA and CB, and ∠ECD lies between CE and CD, so the condition applies. Then AB/ED = CA/CE, so 5/ED = 2/5 and ED = 12.5 cm.
Proof
| Statement | Reason |
|---|---|
| CA/CE = 4/10 = 2/5 | |
| CB/CD = 6/15 = 2/5 | |
| ∠ACB = ∠ECD | vert. opp. ∠s |
| ∴ △CAB ~ △CED | ratio of 2 sides, inc. ∠ |
Common mistakes
These mistakes are in the write-up, not the idea, so they are easy to fix. Check your proof against this list before you move on.
- Writing the letters out of order, such as △APQ ~ △ACB, so that the ratios pair the wrong sides.
- Pairing a side of the small triangle with a piece of a side (PB) instead of the whole side (AB).
- Giving a congruence reason (SSS, SAS, ASA, AAS or RHS). For similar triangles the sides must be in the same ratio, not equal.
- Writing ratio of 2 sides, inc. ∠ when the equal angle is not between the two pairs of sides.
- Listing equal angles without a reason for each one.
Try one
In △ABC, D lies on AB and E lies on AC, with ∠ADE = ∠ACB. AD = 4 cm, AE = 3 cm and AC = 8 cm. Prove that △ADE ~ △ACB, then find AB.
Hint: ∠A is common to both triangles. D matches C, so E matches B.
Answer: ∠DAE = ∠CAB (common ∠) and ∠ADE = ∠ACB (given), so △ADE ~ △ACB (AAA). Then AD/AC = AE/AB, so 4/8 = 3/AB and AB = 6 cm.
More methods and examples for similar triangles
Frequently asked questions
Is AA accepted, or must I write AAA?
The EDB curriculum writes the condition as AAA (or AA), so two pairs of equal angles are enough. In tests, use the abbreviation your school uses; many teachers still expect AAA.
Can I use RHS to prove triangles similar?
No. RHS is a reason for congruent triangles. For two right-angled triangles, use AAA (the right angles and one other pair of equal angles), or find the third sides with Pythagoras' theorem and use 3 sides prop.
Is SSA, two sides and a non-included angle, a reason?
No. Two triangles can have two pairs of sides in the same ratio and an equal angle that is not between them, and still not be similar. The equal angle must be the included angle.
Are congruent triangles similar?
Yes. Congruent triangles are a special case of similar triangles, with every ratio of corresponding sides equal to 1.
Are two quadrilaterals with proportional sides similar?
Not necessarily. A square and a rhombus with no right angle have all sides in the same ratio, but their angles differ, so they are not similar. The three conditions work only for triangles.