Criteria for similarity: AAA, SSS and SAS
Triangles are similar if all angles match (AAA or AA), all sides are proportional (SSS), or one angle matches and its two sides are proportional (SAS).
The criteria at a glance
| Criterion | What you must show | Then |
|---|---|---|
| AAA (angle-angle-angle) | ∠A = ∠P, ∠B = ∠Q and ∠C = ∠R | ΔABC ~ ΔPQR |
| AA (follows from AAA) | Two pairs of angles equal. The third pair is then equal, since the angles of a triangle add to 180° | ΔABC ~ ΔPQR |
| SSS (side-side-side) | AB/PQ = BC/QR = CA/RP | ΔABC ~ ΔPQR |
| SAS (side-angle-side) | ∠A = ∠P and AB/PQ = AC/PR (the angle lies between the two sides) | ΔABC ~ ΔPQR |
Similarity vs congruence tests
For triangles, equal angles force proportional sides, and proportional sides force equal angles. This is not true for quadrilaterals.
How to draw and name similar triangles
- Draw the two triangles side by side in the same position, with matching corners in the same place.
- Mark equal angles with the same sign (one arc, two arcs).
- Write the correspondence in order. If ∠A = ∠P and ∠B = ∠Q, write ΔABC ~ ΔPQR, not ΔABC ~ ΔQPR.
- Read the ratios from the names: AB/PQ = BC/QR = CA/RP.
- If one triangle sits inside the other, redraw them separately before you name them.
Example 1: which criterion?
ΔABC has AB = 3 cm, BC = 4 cm and CA = 5 cm. ΔPQR has PQ = 4.5 cm, QR = 6 cm and RP = 7.5 cm. Are they similar? By which criterion?
- AB/PQ = 3/4.5 = 2/3
- BC/QR = 4/6 = 2/3
- CA/RP = 5/7.5 = 2/3
- All three ratios are equal.
- So ΔABC ~ ΔPQR by the SSS criterion.
Yes, ΔABC ~ ΔPQR (SSS).
Example 2: AA and the right order
In ΔABC, ∠A = 50° and ∠B = 70°. In ΔDEF, ∠D = 50° and ∠F = 60°. Are the triangles similar? Write the correspondence.
- Angle sum of a triangle is 180°.
- ∠C = 180° − 50° − 70° = 60°
- ∠E = 180° − 50° − 60° = 70°
- So ∠A = ∠D = 50°, ∠B = ∠E = 70° and ∠C = ∠F = 60°.
- By AAA (or AA), ΔABC ~ ΔDEF.
Yes. ΔABC ~ ΔDEF by AAA.
Two rangoli or kolam triangles with the same angles but different sizes are similar by AAA. So are a small and a large 30°-60°-90° set square in your geometry box.
1 marks: “In ΔABC and ΔPQR, AB/PQ = AC/PR and ∠A = ∠P. Which criterion makes them similar?” SAS similarity, because the equal angle lies between the two proportional sides.
2 marks: “State the SSS similarity criterion.” If the corresponding sides of two triangles are in the same ratio, their corresponding angles are equal and the triangles are similar.
3 marks: “In ΔABC, D is on AB and E is on AC with AD = 2 cm, AB = 6 cm, AE = 3 cm and AC = 9 cm. Show that ΔADE ~ ΔABC and find DE if BC = 7.5 cm.” AD/AB = 1/3 = AE/AC and ∠A is common, so ΔADE ~ ΔABC (SAS). DE/BC = 1/3, so DE = 2.5 cm.
AAA: angles. SSS: sides. SAS: the angle Sandwiched between two Sides.