Proving Triangles Are Similar
Grade 6The AA condition
Two triangles are similar if two pairs of corresponding angles are equal — this is the AA (angle-angle) condition. Because the angles of a triangle sum to 180°, two equal angles guarantee the third is equal too, so AA is sufficient.
What to write in a proof
Name each pair of equal angles, give the reason (vertically opposite, alternate angles, common angle, angle in a semicircle, etc.), then state "therefore the triangles are similar by AA".
Look for shared angles (a common vertex is a giveaway), vertically opposite angles, and alternate angles in parallel-line diagrams — these are the three most common sources of equal angles in similarity proofs.
Worked example
EXAMPLE
In a triangle, DE is parallel to BC. Prove that triangle ADE is similar to triangle ABC.
Angle A is common to both triangles. Angle ADE = angle ABC (corresponding angles, DE parallel to BC). Two pairs of angles are equal, so triangle ADE is similar to triangle ABC (AA).
