Triangles

Similar triangles have equal angles and sides in the same ratio; the Basic Proportionality Theorem and the AA, SSS and SAS tests decide when triangles are similar.

Key ideas

Similar figures have the same shape; their sizes may differ. All circles are similar, and so are all squares.
Two triangles are similar when their corresponding angles are equal and their corresponding sides are in the same ratio.
Basic Proportionality Theorem (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio.
Converse: if a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
AA: two pairs of equal angles are enough. SSS: all three pairs of sides in the same ratio. SAS: one equal angle with the sides around it in the same ratio.
Write the similarity in matching order: △ABC ~ △PQR means A ↔ P, B ↔ Q, C ↔ R.
Perimeters of similar triangles are in the same ratio as their sides.

Remember

BPT (DE ∥ BC)	AD/DB = AE/EC
△ABC ~ △PQR	AB/PQ = BC/QR = CA/RP
Shadows at the same time	height₁/shadow₁ = height₂/shadow₂

Watch out

Matching the wrong vertices, so the wrong sides are put in a ratio.
Using SAS when the equal angle is not between the two proportional sides.
Writing AD/AB = AE/EC (mixing a part with a whole).

Worked example

Isha's eyes are 1.6 m above the ground. A mirror lies flat on the ground 1 m from Isha's feet, and in it Isha sees the top of a tree that is 5 m from the mirror. How tall is the tree?

Working: the angles at the mirror are equal, so the triangles are similar: height/5 = 1.6/1

Answer: 8 m