Similar Triangles and Proving Similarity
Decide whether two triangles are similar from equal angles or matching side ratios, set the reason out in one line, and use the scale factor to find a missing length.
What a learner can do afterwards
- Show two triangles are similar by naming the pairs of equal angles or by showing three equal side ratios (SSS)
- Match corresponding vertices before writing any ratio
- Find a missing side using the scale factor between matching sides
1 · Read
Two triangles are similar when all three pairs of corresponding angles are equal and the matching sides are in proportion. Like congruent triangles, similar triangles may have scale factor 1, where every matching pair is equal. The angle-angle test is the shortcut: if two pairs of angles match, the third must match too, since the angles of a triangle always sum to 180 degrees. Write the reason in one line, naming the equal pairs.
Before you write any ratio, match the vertices. List them in matching order so each side pairs with its true partner. Shortest goes with shortest and longest with longest. You can also test three pairs of sides directly: triangles are similar by SSS when all three ratios of matching sides are equal. Compare the ratios, not the raw lengths.
Triangle PQR has angles 50, 60 and 70 degrees. Triangle STU has angles 50, 60 and 70 degrees, with S matching P, T matching Q and U matching R. All three angle pairs match, so the triangles are similar by AA. The matching sides are PQ with ST, QR with TU, and PR with SU.
Two similar triangles have a 3 cm side matching a 12 cm side. The scale factor from small to large is 12 divided by 3, which is 4, so a 5 cm side matches 5 times 4, which is 20 cm. Now compare sides 4, 6 and 8 against sides 10, 15 and 20. The ratios are 10/4 = 2.5, 15/6 = 2.5 and 20/8 = 2.5, so these triangles are similar by SSS.
The scale factor has a direction. Choose it for the direction you need: from the smaller triangle to the larger one you multiply by the factor, and going back you divide, or multiply by the reciprocal. Equal angles sit opposite matching sides, which gives a cross-check. If the side between the 50 and 60 degree angles pairs up in both triangles, your pairing is right.
A ratio written from mismatched vertices can give a wrong scale factor and a wrong side, even when the triangles truly are similar. Equal sides can hide the error, so check the pairing. Name the pairs before you calculate. The safe order is: prove similarity by AA or SSS, match the vertices, find the scale factor from one matching pair, then apply it.
Match two angle pairs or three equal side ratios to prove similarity, pair the vertices, then scale. Scale factor 1 means congruent.
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