The big idea Two pairs of congruent angles are enough to prove triangles similar because the third angles must match too.
AA Similarity
If two pairs of corresponding angles are congruent, the triangles are similar. The third pair must match because each triangle's angles add to 180°. That is why the theorem is called AA instead of AAA.
Write ΔABC ~ ΔDEF only when A matches D, B matches E, and C matches F. ΔABC ~ ΔDFE makes a different claim.
All circles are similar, and all squares are similar. Triangles still need a check because being triangle-shaped is not enough.
Fill the missing angle first
For right triangles, the right angles already match. If one acute angle is 48°, its other acute angle is 180 − 90 − 48 = 42°. A second right triangle with a 42° angle has two matching pairs, so AA applies.
Counterexample: one triangle has 46° and 55°, so its third angle is 79°. Another has 46° and 67°. Only one pair matches, so the triangles are not similar.
Same 90°, wrong pairing: compute the missing angles before writing the similarity statement. The labeled 48° may match a 48° that is not labeled in the other triangle.
Scale drawings
A floor plan is a dilation of the room. Corresponding angles stay congruent, so a 38° corner on the page is 38° in the room.
If one square represents 6 inches, a refrigerator measuring 4 by 5 squares is 24 inches by 30 inches. Count the squares, then multiply by the scale. An approximation is reasonable when the source is only a sketch.
Your turn
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