L3 · Similarity criteria: AA, SSS, SAS

Learning goal

“I can decide whether two fully described triangles are similar by checking AA, SSS (all three side ratios equal), or SAS (two side ratios equal around the included angle), state the scale factor, write the similarity statement in correct correspondence, and use the correspondence to find an unknown side.”

Similar — the same shape, possibly a different size: all angles equal in pairs, all sides in the same ratio (congruent is the special case where the ratio is 1).

Scale factor — the single number every side of one triangle is multiplied by to get the corresponding side of the other: from triangle ABC to triangle DEF, scale factor = (a side of DEF) / (the corresponding side of ABC).

Proportional sides — sides whose ratios all agree: AB/DE = BC/EF = CA/FD. One disagreement kills proportionality.

AA criterion — two pairs of equal angles prove similarity, because the angle sum forces the third pair to be equal too.

Worked example

Triangle ABC has angle A = 45 degrees, angle B = 70 degrees (so angle C = 180 - 45 - 70 = 65 degrees). Triangle DEF has angle D = 45 degrees, angle E = 70 degrees (so angle F = 65 degrees). Step 1: Two pairs of equal angles: angle A = angle D, angle B = angle E. Step 2: The third pair needs no measuring: both third angles are 180 degrees minus the same two numbers, so angle C = angle F automatically. Step 3: Same shape guaranteed: triangle ABC is similar to triangle DEF, by AA. Say it plainly: angles are the shape of a triangle. Two matching angle pairs - with the third forced by the 180-degree sum - mean the shapes are identical, whatever the sizes. That is all AA claims, and it is enough.

Check yourself — explain it

AA needs only TWO angle pairs because...

Paired problem — example still visible

Triangle GHI has angle G = 30 degrees, angle H = 90 degrees. Triangle JKL has angle J = 90 degrees, angle L = 60 degrees. Are they similar? Write the statement. -> Angle I = 180 - 30 - 90 = 60 degrees = angle L, and angle H = angle J = 90 degrees, so AA applies. Correspondence: G (30 degrees) matches K (30 degrees), H matches J, I matches L: triangle GHI is similar to triangle KJL

Faded completion — last step is yours

"Two pairs of equal angles prove similarity by the ___ criterion" -> AA

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