Skill: similar-triangles

Similar Triangles & Proportions

Similar means same shape, proportional sides. Prove it with AA, SSS~, or SAS~ — then solve with proportions and measure the unmeasurable, like a tree’s height from its shadow.

1 Key ideas

Three ways to prove similarity, one scale factor, and proportions that must keep the order.

AA Angle-Angle

Two pairs of congruent angles. The third pair follows automatically (angle sum), so two is enough.

SSS~ proportional sides

All three side ratios match: 3/6 = 4/8 = 5/10. Same shape, scaled.

SAS~ sides + included angle

Two pairs of sides in the same ratio, with the included angles congruent.

Example: AB/DE = AC/DF = 2 and ∠A ≅ ∠D.

△ABC ~ △DEF ⇒ AB/DE = BC/EF = AC/DF = k (the scale factor)

Order matters: match smallest to smallest, following the similarity statement. In △ABC ~ △DEF, side AB corresponds to DE — not to DF or EF. Wrong order, wrong proportion.

Similar ≠ congruent. Similar triangles have the same shape with proportional (not equal) sides. Congruent is the special case where the scale factor k = 1.

Quick check 1 — ∠A ≅ ∠D and ∠B ≅ ∠E. Which similarity criterion applies?

Quick check 2 — △ABC ~ △DEF. AB = 6, DE = 9, BC = 8. Find EF.

Quick check 3 — △ABC ~ △DEF with scale factor 3 from △ABC to △DEF. AB = 5. Find DE.

Quick check 4 — Triangle sides 3, 4, 5 and 6, 8, 10. Are the triangles similar?

2 Worked examples

Solving a proportion, then the classic shadow problem.

Example 1 — solve the proportion

△ABC ~ △DEF. AB = 4, DE = 10, BC = 6. Find EF.

Write the proportion in order: AB/DE = BC/EF.
Substitute: 4/10 = 6/EF.
Cross-multiply: 4 × EF = 60, so EF = 15.

Example 2 — indirect measurement

A person 6 ft tall casts a 4 ft shadow. At the same time, a tree casts a 20 ft shadow. How tall is the tree?

The sun’s rays make the same angle for both, so the two triangles are similar by AA.
Proportion: person/shadow = tree/shadow → 6/4 = h/20.
Cross-multiply: 4h = 120, so h = 30 ft.

Quick check 5 — A 5 ft person casts a 3 ft shadow; a flagpole casts a 15 ft shadow. The flagpole is…

Quick check 6 — △ABC ~ △DEF. Complete the proportion: AB/DE = BC / ___

Practice proportions Open the proportion solver