SSS (Side-Side-Side) Congruence
If all three sides of one triangle are congruent to all three sides of another, the triangles must be congruent. You don't need to check the angles — the sides force them.
Introduction
Checking all 6 parts (3 sides + 3 angles) is tedious. SSS is the first shortcut: if you can show three pairs of sides match, you're done. This works because three fixed side lengths can form exactly one triangle shape.
Past Knowledge
Congruence definition (5.2.1). Triangle Inequality. Rigid motions (Ch 4).
Today's Goal
State, justify, and apply the SSS Congruence Postulate.
Future Success
SAS (5.2.3), ASA (5.2.4), CPCTC proofs (5.2.7).
Key Concepts
SSS Congruence Postulate
If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
If , then .
Why It Works — Rigidity
If you build a triangle out of three sticks of fixed lengths, the triangle is rigid: you cannot change its angles without breaking a stick. This is why triangles appear in bridges, roofs, and bicycles — they don't flex. It also means three side lengths determine a unique triangle shape.
Theorem & Proof
Two-Column Proof: SSS Congruence (Rigid Motion Argument)
Given: and with
Prove:
Strategy: Use rigid motions from Chapter 4 to map one triangle onto the other.
Step through the rigid motion argument: align one side, then the third vertex is forced by two-circle intersection.
| # | Statement | Reason |
|---|---|---|
| 1 | Given | |
| 2 | Apply a rigid motion (translation + rotation) to map and | Since , a translation followed by a rotation can align these two sides exactly (rigid motions preserve length) |
| 3 | maps to a point such that and | Rigid motions preserve all distances () |
| 4 | lies on the circle of radius centered at AND on the circle of radius centered at | Definition of circle (set of points at a given distance from center) |
| 5 | These two circles intersect in at most 2 points, on opposite sides of | Two distinct circles can intersect in at most 2 points |
| 6 | If , we are done. If , reflect across to map | and are symmetric about (equidistant from both and ); reflection is a rigid motion |
| 7 | A sequence of rigid motions maps onto exactly; congruence by definition (Ch 4) |
∎ If three pairs of sides are congruent, the triangles are congruent.
Worked Examples
Direct SSS Application
Given with sides 8, 6, 10 and with sides 10, 8, 6. Are they congruent by SSS?
Match corresponding sides: ✓
The same three lengths appear in both triangles.
Yes, by SSS. (Match by length: )
Using Algebra with SSS
In and : , , , , , . Find so triangles are congruent by SSS.
Two pairs already match: and .
For SSS, we need :
Check: ✓
. by SSS.
SSS with a Shared Side
In quadrilateral , diagonal divides it into and . Given and . Prove .
Side 1: (given)
Side 2: (given)
Side 3: (Reflexive Property — shared side)
All three pairs of sides are congruent.
by SSS.
Common Pitfalls
Forgetting the Reflexive Property
When two triangles share a side, that shared side is congruent to itself. Always cite the Reflexive Property — this is one of the most common SSS “free” sides.
Matching Sides Incorrectly
Corresponding sides must connect corresponding vertices. If , then corresponds to (not or ).
Real-Life Applications
Bridge Engineering — Truss Rigidity
Triangular trusses in bridges are rigid precisely because of SSS: once the three member lengths are fixed, the triangle cannot deform. Quadrilaterals, by contrast, can flex (think of a parallelogram becoming a rhombus). That's why engineers triangulate every structure.
3D Printing — Mesh Verification
3D printers convert models into triangular meshes. The slicing software uses SSS to verify that corresponding triangles in mirrored or duplicated parts are congruent, ensuring the printed object matches the digital design.
Practice Quiz
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