Two equal angles are enough: the third follows, and the triangles are similar.
Key formulas
Hunt equal angles
Shared angles, vertically opposite angles, and parallel-line angles are the usual sources.
Write the correspondence
If ∠A=∠D and ∠B=∠E, write △ABC ∼ △DEF — order matters for the side ratios.
Ratio
AB/DE = BC/EF = CA/FD. Cross-multiply carefully.
Worked example
Worked example
Problem. Two triangles share ∠A. Each also has a right angle. △ABC has AB = 5 opposite the other acute angle matching △DEF with DE = 10. If BC = 8, find EF.
Solution. AA similarity holds. Scale factor 10/5 = 2. Corresponding side EF = 8 × 2 = 16.
Shared angle
Two triangles sharing ∠A and both having a right angle are similar by AA.
Review quiz
Sharing ∠A and both having ∠=70° elsewhere is enough for similarity by:
Practice questions
- Name two ways to get equal angles without being told numbers.
- △ABC ∼ △XYZ with AB=6, XY=9, BC=8. Find YZ.
Study notes
Is AAA different from AA?
AAA is automatic once AA holds, because angle sum is 180°.
More in Plane Geometry
- Circle tangents: the radius is perpendicular
- Right-triangle lengths: Pythagoras and altitude
- Similar triangles: matching sides in proportion
- Chord theorem: perpendicular from the centre
- Isosceles triangles and equal base angles
- Alternate segment theorem with a short proof
- Converse of Pythagoras: proving a right angle
- Heron’s formula when height is hidden