Similarity of triangles

For triangles, a quicker test than checking everything is enough. If two angles of one triangle equal two angles of another, the third angles are equal too, because the angles of a triangle add to 180°. The triangles then have the same shape: they are similar, and their corresponding sides are proportional. This is the AA case. It often appears with parallel lines. In a triangle ABC, a line DE parallel to BC (D on AB, E on AC) makes ∠ADE = ∠ABC and ∠AED = ∠ACB (corresponding angles) and shares angle A, so △ADE ~ △ABC. When two lines AC and BD cross at E and AB ∥ CD, the alternate angles are equal and the vertical angles at E are equal, so △ABE ~ △CDE. Write the similarity with the vertices in matching order, then use the equal ratios.