Triangles
Similarity criteria
Use AA, SSS and SAS to tell whether two triangles are similar.
For two triangles, similar means corresponding angles equal and corresponding sides in the same ratio. Luckily you never have to check all six facts. Each of these criteria is enough on its own:
| Criterion | What you must show |
|---|---|
| AAA (or just AA) | two angles of one equal two angles of the other (the third then matches too, since angles add to 180°) |
| SSS | all three pairs of sides are in the same ratio |
| SAS | one angle is equal, and the two sides forming that angle are in the same ratio |
Careful with SAS: the angle must be the one between the two sides. Two sides and some other angle (SSA) is not a criterion.
The order of the letters matters. ΔABC ~ ΔDEF means A matches D, B matches E and C matches F. Mark equal angles with one, two and three arcs, then walk the same path in both triangles.
Matching sides by size (SSS): ΔABC has sides 2, 2.5, 3 and ΔPQR has sides 4, 5, 6. Put each in order from smallest to biggest: 2/4 = 2.5/5 = 3/6 = ½. They are similar. Then match each vertex to its partner: ΔABC ~ ΔQRP.
A routine for proofs:
- First, name the criterion and give the reasons (an angle that is given, a common angle, vertically opposite angles, angles from parallel lines).
- Then write the similarity with the letters in the correct order.
- Last, use it: corresponding sides of similar triangles are in proportion.
Why triangles are special: for a quadrilateral, equal angles alone are not enough (think of a square and a rectangle). For triangles, equal angles force the sides into the same ratio, and the other way round.
Practise this lesson
Questions that get harder as you improve, a Tutor that helps without giving the answer away, and progress you can see. Free for every learner.
No account, no email address. Opens the real product.