Geometry A hub · 03.03

Triangles and Similarity

Two matching angles is enough. The third is automatic. Name the letters in order.

AA Similarity

If two pairs of corresponding angles are congruent, the triangles are similar. The third pair must match too, because each triangle's angles add to 180°. That is why the course does not call it AAA: the third A is free.

Write ΔABC ~ ΔDEF only if A matches D, B matches E, C matches F. ΔABC ~ ΔDFE is a different claim.

All circles are similar. All squares are similar. Triangles need a check, because "triangle-shaped" is not enough.

Fill the missing angle first

Right triangles: the right angles already match. If one acute angle is 48° and the other triangle shows 42° in a different corner, compute 180 − 90 − 48 = 42. Then you have two matches, so AA.

Counterexample: 46° and 55° in one triangle (third is 79°) against 46° and 67° in the other. Only one pair matches. Not similar.

Same 90°, wrong pairing: a 48° in the small triangle might correspond to a 48° you have to compute, not to the 42° that is already labeled. Compute before you name the similarity statement.

Scale drawings

A floor plan is a dilation of the room. Corresponding angles stay, so a 38° corner on the page is 38° in the kitchen. You only need two pairs of matching angles to name the triangles similar.

If 1 square = 6 inches, a fridge that is 4 by 5 squares is 24 in. by 30 in. Count squares, then multiply by the scale. Approximate is allowed when the drawing is a sketch.

Your turn