SSS Similarity
If all three pairs of corresponding sides are proportional (same ratio), then the triangles are similar — no angle information needed.
Introduction
Sometimes you know all three sides of both triangles but no angles. The SSS Similarity Theorem says: if all three ratios of corresponding sides are equal, the triangles must be similar.
Past Knowledge
AA Similarity (7.2.1). Proportions (7.1.1–7.1.2). SSS Congruence (5.2.2).
Today's Goal
Determine similarity using three pairs of side lengths.
Future Success
SAS Similarity (7.2.3), perimeters & areas (7.3.3).
Key Concepts
SSS Similarity Theorem
If the three pairs of corresponding sides of two triangles are proportional:
then .
How to Check
- List the sides of each triangle from shortest to longest
- Pair them: shortest↔shortest, middle↔middle, longest↔longest
- Compute all three ratios — if they're all equal, the triangles are similar
Theorem & Proof
Two-Column Proof: SSS Similarity Theorem
Given:
Prove:
Strategy: Construct a dilation of that makes its sides equal to 's sides, then use SSS Congruence.
| # | Statement | Reason |
|---|---|---|
| 1 | Given (let the common ratio be ) | |
| 2 | Apply a dilation centered at with scale factor to , producing | A dilation exists for every center and scale factor |
| 3 | , , | Dilation multiplies all lengths by ; substituting from step 1 |
| 4 | SSS Congruence (all three pairs of sides are equal by step 3) | |
| 5 | A dilation produces a similar figure (definition of similarity) | |
| 6 | Steps 4–5: , so |
∎ If all three pairs of sides share the same ratio, the triangles are similar.
Worked Examples
Checking SSS Similarity
: sides 4, 6, 8. : sides 6, 9, 12. Are they similar?
Pair shortest to shortest:
Middle:
Longest:
All ratios equal → SSS Similarity.
Yes, by SSS Similarity (scale factor )
Not Similar
: sides 3, 5, 7. : sides 6, 10, 15. Similar?
, ,
The ratios are NOT all equal.
Not similar — the third ratio doesn't match.
Finding a Missing Side for Similarity
: 5, 8, 10 and : 7.5, 12, . Find so the triangles are similar.
Scale factor: and ✓
For similarity: →
Common Pitfalls
Pairing Sides Incorrectly
Always pair shortest↔shortest, not arbitrary sides. If the order doesn't match, the ratios won't be equal even if the triangles are similar.
Checking Only Two of Three Ratios
Unlike AA (where 2 is enough), SSS requires all three ratios to be equal. Two matching ratios with one off is NOT similar.
Real-Life Applications
Quality Control
Manufacturers check that machined parts are similar to the blueprint by measuring all dimensions and verifying the ratios. If all ratios match the scale factor, the part is correctly made.
Photo Printing
A 4×6 photo is similar to an 8×12 print (ratio 1:2), but NOT similar to a 5×7 print (). This is why 5×7 prints are slightly cropped.
Practice Quiz
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