Lesson 6 — SSS Congruence
Big idea: If all three sides of one triangle match all three sides of another, the triangles are congruent. Three side lengths completely determine a triangle.
Words to know
| Word | Meaning |
|---|---|
| SSS | Side-Side-Side: all three pairs of corresponding sides are equal. |
| Rigid | Cannot change shape; the shape is fixed by the side lengths. |
| Truss | A framework of triangles used to strengthen a structure. |
| Corresponding parts | The matching sides and angles of congruent triangles. |
Concepts to know
- SSS Theorem: If the three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent.
- A triangle is rigid: its three side lengths force its angles.
- A quadrilateral is not rigid: four side lengths can still wobble into different shapes.
- Bridges and roofs use triangles because triangles do not collapse.
- After proving triangles congruent by SSS, use CPCTC for matching angles or sides.
- SSS needs all three pairs of sides.
The proof pattern
- Show three pairs of corresponding sides are equal.
- State SSS.
- Use CPCTC for the needed angle or side.
Common sources of equal sides
| Source | Equal sides |
|---|---|
| Given | As stated |
| Shared side | Reflexive property |
| Midpoint | Two equal halves |
| Isosceles triangle | Two equal sides |
| Linkage or rods | Equal lengths by construction |
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Prove two triangles congruent | Mark three pairs of equal sides, then state SSS. |
| Prove an angle equal | Prove the triangles congruent by SSS, then use CPCTC. |
| Explain why a structure is rigid | Point to the triangles; their side lengths cannot change. |
| Explain why a quadrilateral wobbles | Four sides do not fix the angles. |
| Use a shared side | Cite the reflexive property. |
| Use a midpoint | Write the two equal segments first. |
Worked example
Plumb level: AB = AC, BW = CW, and AW is shared.
Prove: ∠1 = ∠2.
| Statement | Reason |
|---|---|
| 1. AB = AC. | Given |
| 2. BW = CW. | Given |
| 3. AW = AW. | Reflexive property |
| 4. ΔABW ≅ ΔACW. | SSS |
| 5. ∠1 = ∠2. | CPCTC |
Once ∠1 = ∠2 and they form a linear pair, each is 90°, so AW is perpendicular to BC.
Watch out
- SSS requires three pairs of sides, not two.
- Same perimeter does not mean same side lengths.
- Do not confuse SSS with SAS. SAS needs an included angle.
- Keep the correspondence order correct when naming the triangles.