Two triangles with two pairs of equal angles are always similar!
Two triangles are SIMILAR if they have the same shape but not necessarily the same size — corresponding angles are equal and corresponding sides are proportional. The Angle-Angle (AA) Criterion: if two angles of one triangle equal two angles of another, the triangles must be similar (the third angle is forced to match because all angles sum to 180°).
AA Criterion: Two pairs of equal angles → triangles are similar. Notation: △ABC ∼ △DEF.
A 6 ft person casts a 4 ft shadow. A nearby tree casts a 20 ft shadow. Both form right triangles with the ground. Two matching angles: right angle (90°) + sun angle. AA criterion satisfied! Proportion: 6/4 = tree height/20 Tree height = 6 × 20 ÷ 4 = 30 ft
Using similar triangles to find unknown lengths:
Triangle PQR has angles 40°, 60°, 80°. Triangle STU has angles 40°, 80°, 60°. Are they similar?
Remember This!
Parallel lines cut by a transversal create equal corresponding angles — a classic setup for AA similarity proofs. Look for parallel sides in overlapping triangles!